SEU Statistics State Hypothesis Worksheet

Section 9.2(Problem 8) State hypothesis H0,H1 and draw conclusion

Section 9.3(Problem 1,9)

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Section 9.4(Problem 1 only a and b)

Section 9.5(Problem 1 part a)

Problem 8 on ch.9 supplement exercises

Reserve Problem Chapter 9 Section 2 Problem 8

Humans are known to have a mean gestation period of 280 days

(from last menstruation) with a standard deviation of about 9 days.

A hospital wondered whether there was any evidence that their

patients were at risk for giving birth prematurely. In a random

sample of 70 women, the average gestation time was 274.7 days.

a. Is the alternative hypothesis one- or two-sided?

b. What is the P-value of the test statistic?

Reserve Problem Chapter 9 Section 3 Problem 1

We have 5-year statistics of the average amount of wheat crop

(tons) harvested from 1 km2 per year, the results are as follows:

560, 525, 496, 543, 499. Test the hypothesis that the mean wheat

crop is 550 tons per 1 km2 per year (? = 0.05) and choose the correct

answer. Determine a 95% confidence interval on the mean

wheat crop. Determine whether the hypothesis that the mean

wheat crop is 550 tons per 1 km2 per year (? = 0.05) is true based

on the 95% confidence interval?

Reserve Problem Chapter 9 Section 3 Problem 9

Human oral normal body temperature is believed to be 98.6? F,

but there is evidence that it actually should be 98.2? F

[Mackowiak, Wasserman, Steven and Levine, JAMA (1992, Vol. 268(12), pp. 1578–1580)].

From a sample of 52 healthy

adults, the mean oral temperature was 98.285 with a standard

deviation of 0.625 degrees.

a. What are the null and alternative hypotheses?

b. Test the null hypothesis at ? = 0.05. Find t0. Is it possible to

reject H0 hypothesis at the 0.05 level of significance?

c. Construct a 95% confidence interval to answer the same

question.

Reserve Problem Chapter 9 Section 4 Problem 1

A group of 15 students has performed an experiment, they measured

the coefficient of thermal expansion for aluminum. The

results are as follows (10?6 K?1):22.0, 25.9, 25.6, 23.1, 22.7, 25.6,

24.9, 21.9, 26.1, 24.3, 23.5, 20.7, 21.4, 23.5, 20.4.

a. Is there strong evidence to conclude that the standard deviation

in this experiment exceeds 3? Use ? = 0.05.

b. Find the P-value for this test.

Reserve Problem Chapter 9 Section 5 Problem 1

A manufacturer of soap bubble liquid will test a new solution

formula. The solution will be approved, if the percent of produced

parisons, in which the content does not allow the bubbles

to inflate, does not exceed 7%. A random sample of 700 parisons

contains 50 defective parisons. Use the z-values rounded to three

decimal places to obtain the answers.

a. Formulate and test an appropriate set of hypotheses to determine

whether the solution can be approved. Use ? = 0.05.

Find the P-value.

Reserve Supplemental Exercises Chapter 9 Problem 8

A communication channel is being monitored by recording the

number of errors in a string of 1000 bits. Data for 20 of these

strings follow:

3 1 0 1 3 2 4 1 3 1

1 1 2 3 3 2 0 2 0 1

Consider the 20 observations collected on the number of errors in

a string of 1000 bits of a communication channel.

a. Based on the description of the random variable and these

20 observations, is a binomial distribution an appropriate

model? Perform a goodness-of-fit procedure with ? = 0.05.

Let the expected frequencies be at least three.

b. Calculate the P-value for this test.

Section 9.2(Problem 8) State hypothesis H0,H1 and draw conclusion
Section 9.3(Problem 1,9)
Section 9.4(Problem 1 only a and b)
Section 9.5(Problem 1 part a)
Problem 8 on ch.9 supplement exercises
Reserve Problem Chapter 9 Section 2 Problem 8
Humans are known to have a mean gestation period of 280 days
(from last menstruation) with a standard deviation of about 9 days.
A hospital wondered whether there was any evidence that their
patients were at risk for giving birth prematurely. In a random
sample of 70 women, the average gestation time was 274.7 days.
a. Is the alternative hypothesis one- or two-sided?
b. What is the P-value of the test statistic?
Reserve Problem Chapter 9 Section 3 Problem 1
We have 5-year statistics of the average amount of wheat crop
(tons) harvested from 1 km2 per year, the results are as follows:
560, 525, 496, 543, 499. Test the hypothesis that the mean wheat
crop is 550 tons per 1 km2 per year (α = 0.05) and choose the correct
answer. Determine a 95% confidence interval on the mean
wheat crop. Determine whether the hypothesis that the mean
wheat crop is 550 tons per 1 km2 per year (α = 0.05) is true based
on the 95% confidence interval?
Reserve Problem Chapter 9 Section 3 Problem 9
Human oral normal body temperature is believed to be 98.6∘ F,
but there is evidence that it actually should be 98.2∘ F
[Mackowiak, Wasserman, Steven and Levine, JAMA (1992, Vol. 268(12), pp. 1578–1580)].
From a sample of 52 healthy
adults, the mean oral temperature was 98.285 with a standard
deviation of 0.625 degrees.
a. What are the null and alternative hypotheses?
b. Test the null hypothesis at α = 0.05. Find t0. Is it possible to
reject H0 hypothesis at the 0.05 level of significance?
c. Construct a 95% confidence interval to answer the same
question.
Reserve Problem Chapter 9 Section 4 Problem 1
A group of 15 students has performed an experiment, they measured
the coefficient of thermal expansion for aluminum. The
results are as follows (10−6 K−1):22.0, 25.9, 25.6, 23.1, 22.7, 25.6,
24.9, 21.9, 26.1, 24.3, 23.5, 20.7, 21.4, 23.5, 20.4.
a. Is there strong evidence to conclude that the standard deviation
in this experiment exceeds 3? Use α = 0.05.
b. Find the P-value for this test.
Reserve Problem Chapter 9 Section 5 Problem 1
A manufacturer of soap bubble liquid will test a new solution
formula. The solution will be approved, if the percent of produced
parisons, in which the content does not allow the bubbles
to inflate, does not exceed 7%. A random sample of 700 parisons
contains 50 defective parisons. Use the z-values rounded to three
decimal places to obtain the answers.
a. Formulate and test an appropriate set of hypotheses to determine
whether the solution can be approved. Use α = 0.05.
Find the P-value.
Reserve Supplemental Exercises Chapter 9 Problem 8
A communication channel is being monitored by recording the
number of errors in a string of 1000 bits. Data for 20 of these
strings follow:
3101324131
1123320201
Consider the 20 observations collected on the number of errors in
a string of 1000 bits of a communication channel.
a. Based on the description of the random variable and these
20 observations, is a binomial distribution an appropriate
model? Perform a goodness-of-fit procedure with α = 0.05.
Let the expected frequencies be at least three.
b. Calculate the P-value for this test.
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Reserve Problems
The following problems have been reserved for your use in assignments and testing and do not
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WP Problem available in WileyPLUS at instructor’s discretion.
Chapter 2 Exercises
Exercises for Section 2.1
k
. WP Reserve Problem Chapter 2 Section 1 Problem 1
Determine the sample space for the following random experiment. It is known that the delivery time is within 10 to 46 hours.
Consider an experiment that records the time from an order until
the shipment arrives.
. WP Reserve Problem Chapter 2 Section 1 Problem 2
Consider an experiment that records the number of bushels (and
fractions of bushels) of corn produced in an acre. The yield is
assumed to be between 130 and 230 bushels. Determine the corresponding sample space.
. WP Reserve Problem Chapter 2 Section 1 Problem 3
Consider an experiment that records the error (diﬀerence)
between the actual and estimated quarterly revenue of a corporation. Determine the corresponding sample space.
. WP Reserve Problem Chapter 2 Section 1 Problem 4
A conceptual smartphone design uses a parachute technique to
avoid severe damage to the screen and phone-case. Each of four
nozzles located on the case might be either functional or defective
after a year.
a. Describe the sample space in terms of the condition (functional or defective) of each nozzle after a year. Let “F”
denote a functional nozzle after a year and “D” denote a
defective one.
b. How many outcomes are in the event deﬁned by two defective nozzles?
. WP Reserve Problem Chapter 2 Section 1 Problem 5
Let X denote the grams of gold obtained in a ton of ore. Consider
the two events A = {x|0 ≤ x < 3} and B = {x|2 < x < 4}. Determine the following events. a. A ∩ B b. A ∪ B d. A′ ∩ B′ c. A′ . WP Reserve Problem Chapter 2 Section 1 Problem 6 Each of three machined parts is classiﬁed as either above or below the target speciﬁcation for the part. Let a and b denote a part above and below the speciﬁcation, respectively. Provide a reasonable description of the sample space for this random experiment. k . WP Reserve Problem Chapter 2 Section 1 Problem 7 Each of 24 Web sites is classiﬁed as containing or not containing banner ads. Provide a reasonable description of the sample space for this random experiment. . WP Reserve Problem Chapter 2 Section 1 Problem 8 A scale that displays two decimal places is used to measure material feeds in a chemical plant in tons. Choose a reasonable description of the sample space for this random experiment. . WP Reserve Problem Chapter 2 Section 1 Problem 9 Provide a reasonable description of the sample space for a measurement of the concentration of ozone to the nearest part per billion. . WP Reserve Problem Chapter 2 Section 1 Problem 10 The time of a chemical reaction is recorded to the nearest millisecond. Provide a reasonable description of the sample space for this experiment. . WP Reserve Problem Chapter 2 Section 1 Problem 11 An order for a computer system can specify memory of 4, 8, or 12 gigabytes and disk storage of 200, 300, or 400 gigabytes. Choose the diagrams that describe the set of possible orders. . WP Reserve Problem Chapter 2 Section 1 Problem 12 Three events are shown on the Venn diagram in the following ﬁgure: A B C Match the ﬁgures and the corresponding events. k Trim Size: 8in x 10in STD-H8x10 R-2 bend_RP.tex V1 - 11/08/2017 8:50pm Page R-2 Reserve Problems . WP Reserve Problem Chapter 2 Section 1 Problem 13 Samples of a cast aluminum part are classiﬁed on the basis of surface ﬁnish (in microinches) and edge ﬁnish. The results of 100 parts are summarized as follows: Edge Finish Excellent Surface Finish Good Excellent 70 1 Good 12 17 Let A denote the event that a sample has excellent surface ﬁnish, and let B denote the event that a sample has excellent edge ﬁnish. Determine the number of samples in each of the following events. . WP Reserve Problem Chapter 2 Section 1 Problem 14 Counts of the Web pages provided by each of two computer servers in a selected hour of the day are recorded. Let A denote the event that at least 10 pages are provided by server 1, and let B denote the event that at least 20 pages are provided by server 2. For each of the following events choose the corresponding sample space graph: a. A b. B c. A ∩ B d. A ∪ B k Montgomery7e k . WP Reserve Problem Chapter 2 Section 1 Problem 15 A reactor’s rise time is measured in minutes (and fractions of minutes). Let the sample space for the rise time of each batch be positive, real numbers. Consider the rise times of two batches. Let A denote the event that the rise time of batch 1 is less than 72.5 minutes, and let B denote the event that the rise time of batch 2 is greater than 52.5 minutes. For each of the following events choose the corresponding sample space graph: a. A b. B’ c. A ∩ B d. A ∪ B . WP Reserve Problem Chapter 2 Section 1 Problem 16 A byte is a sequence of eight bits and each bit is either 0 or 1. a. How many diﬀerent bytes are possible? b. If the ﬁrst bit of a byte is a parity check, that is, the ﬁrst byte is determined from the other seven bits, how many diﬀerent bytes are possible? . WP Reserve Problem Chapter 2 Section 1 Problem 17 In a chemical plant, 24 holding tanks are used for ﬁnal product storage. Four tanks are selected at random and without replacement. Suppose that four of the tanks contain material in which the viscosity exceeds the customer requirements. a. What is the probability that exactly one tank in the sample contains high-viscosity material? b. What is the probability that at least one tank in the sample contains high-viscosity material? c. In addition to the four tanks with high-viscosity levels, four diﬀerent tanks contain material with high impurities. What is the probability that exactly one tank in the sample contains high-viscosity material and exactly one tank in the sample contains material with high impurities? . WP Reserve Problem Chapter 2 Section 1 Problem 18 An article in The Journal of Data Science [“A Statistical Analysis of Well Failures in Baltimore County” (2009, Vol. 7, pp. 111-127)] provided the following table of well failures for diﬀerent geological formation groups in Baltimore County. Wells Geological Formation Group Failed Total Gneiss 170 1685 Granite 2 28 Loch raven schist 443 3733 Maﬁc 14 363 Marble 29 309 Prettyboy schist 60 1403 Other schists 46 933 Serpentine 3 39 k Let A denote the event that the geological formation has more than 1000 wells, and let B denote the event that a well failed. Determine the number of wells in each of the following events. a. A ∩ B b. A′ c. A ∪ B ′ e. A′ ∩ B′ d. A ∪ B Exercises for Section 2.2 . WP Reserve Problem Chapter 2 Section 2 Problem 1 Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of orders in the previous month. The number of customers in each category are shown in the following table. Customers are to be selected without replacement. Shipping Contract Express Standard Total No orders 25 15 40 One order 65 45 110 More than one order 40 20 60 k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems a. How many diﬀerent samples of eight express customers are possible? b. How many diﬀerent samples of ten standard customers are possible? c. How many diﬀerent samples of eight express and ten standard customers are possible? . WP Reserve Problem Chapter 2 Section 2 Problem 2 Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of orders in the previous month. The number of customers in each category are shown in the following table. Customers are to be selected without replacement. Shipping Contract k Express Standard Total No orders 25 15 40 One order 65 45 110 More than one order 40 20 60 a. Suppose that 10 express customers are selected without replacement. How many samples contain exactly one customer with more than one order last month? b. Suppose that 10 express customers are selected without replacement. How many samples contain at least one customer with more than one order last month? c. Suppose that 10 express and 15 standard customers are selected without replacement. How many samples contain exactly one express customer and exactly one standard customer with no orders last month? 8:50pm Page R-3 R-3 . WP Reserve Problem Chapter 2 Section 2 Problem 3 A committee will be formed with 4 managers and 3 engineers selected without replacement from 10 managers and 20 engineers. How many diﬀerent committees are possible? . WP Reserve Problem Chapter 2 Section 2 Problem 4 Code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (ﬁve bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white) space appears between each bar. The original speciﬁcation (since revised) used exactly two wide bars and one wide space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). One character is held back as a start and stop delimiter. For each of the following cases, how many characters can be encoded? a. The constraint of exactly two wide bars is replaced with one that requires exactly one wide bar. b. The constraint of exactly two wide bars is replaced with one that allows either one or two wide bars. c. The constraint of exactly two wide bars is dropped. d. The constraints of exactly two wide bars and one wide space are dropped. k Exercises for Section 2.3 . WP Reserve Problem Chapter 2 Section 3 Problem 1 In a car racing competition, four car brands are competing, each having two cars representing them. Cars from the same company have the same probability of winning. Each of the cars from Acura, Alfa Romeo and Audi has the same probability of winning the match, while each of Aston Martin’s cars has the probability of winning 1.57 times the probability of winning for each of the other brands’ cars. a. What is the chance of winning for each of the Aston Martin’s cars? What is the chance of winning for each other cars? b. What is the probability of winning for Aston Martin brand? What is the probability of winning for each other brand? . WP Reserve Problem Chapter 2 Section 3 Problem 2 Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of orders in the previous month. The number of customers in each category are shown in the following table. k Shipping Contract Express Standard Total No orders 25 15 40 One order 65 44 109 More than one order 44 20 64 Total 134 79 213 Suppose that three customers are selected randomly, without replacement, for a survey. a. What is the probability that at least two selected customers use express shipping? b. What is the probability that customers from all three order categories are selected? . WP Reserve Problem Chapter 2 Section 3 Problem 3 A committee will be formed with 2 managers and 4 engineers selected randomly without replacement from 9 managers and 18 engineers. Trim Size: 8in x 10in STD-H8x10 R-4 k bend_RP.tex V1 - 11/08/2017 8:50pm Page R-4 Reserve Problems a. What is the probability that the speciﬁc engineer Jane and the speciﬁc manager Mary are on the committee? b. Each of the engineers diﬀer in years of experience. What is the probability that the most experienced and least experienced engineers are on the committee? . WP Reserve Problem Chapter 2 Section 3 Problem 4 Each of the possible ﬁve outcomes of a random experiment is equally likely. The sample space is [a, b, c, d, e]. Let A denote the event [a, b], and let B denote the event [c, d, e]. Determine the following: a. P(A) = b. P(B) = c. P(A′ ) = d. P(A ∪ B) = e. P(A ∩ B) = WP Reserve Problem Chapter 2 Section 3 Problem 5 . The following table summarizes 186 endothermic reactions involving sodium bicarbonate. Final Temperature Conditions k Montgomery7e Heat absorbed (cal) Below Target Above Target 266 K 12 32 271 K 44 16 274 K 46 36 Let A denote the event that a reaction’s ﬁnal temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Determine the following probabilities. a. P(A ∩ B) = b. P(A′ ) = c. P(A ∪ B) = d. P(A ∪ B′ ) = e. P(A′ ∩ B′ ) = . WP Reserve Problem Chapter 2 Section 3 Problem 6 An article in The Journal of Data Science [“A Statistical Analysis of Well Failures in Baltimore County” (2009, Vol. 7, pp. 111-127)] provided the following table of well failures for diﬀerent geological formation groups in Baltimore County. Wells Geological Formation Group Failed Total Gneiss Granite Loch raven schist Maﬁc Marble Prettyboy schist Other schists Serpentine 170 2 443 14 29 60 46 3 1685 28 3733 363 309 1403 933 39 Let A denote the event that the geological formation has more than 1000 wells, and let B denote the event that a well failed. Determine the following probabilities. a. P(A ∩ B) = b. P(A′ ) = c. P(A ∪ B) = d. P(A ∪ B′ ) = e. P(A′ ∩ B′ ) = . WP Reserve Problem Chapter 2 Section 3 Problem 7 Code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (ﬁve bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white) space appears between each bar. The original speciﬁcation (since revised) used exactly two wide bars and one wide space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally likely (none is held back as a delimiter). Determine the probability for each of the following: a. A wide space occurs before a narrow space. b. Two wide bars occur consecutively. c. Two consecutive wide bars are at the start or end. d. The middle bar is wide. Exercises for Section 2.4 . WP Reserve Problem Chapter 2 Section 4 Problem 1 Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of orders in the previous month. The number of customers in each category are shown in the following table. Customers are to be selected without replacement. Shipping Contract Express Standard Total No orders One order More than one order 22 64 44 17 44 21 39 108 65 a. What is the probability that a randomly selected customer had at least one order in the previous month or uses express shipping? b. What is the probability that a randomly selected customer uses standard shipping or had no orders? . WP Reserve Problem Chapter 2 Section 4 Problem 2 A committee will be formed with 4 managers and 4 engineers selected randomly without replacement from 11 managers and 16 engineers. What is the probability that engineer Jane or manager Mary is on the committee? k k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems . WP Reserve Problem Chapter 2 Section 4 Problem 3 Consider the hospital emergency department visits shown in table below. People may leave without being seen by a physician, and those visits are denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may not be admitted for a stay in the hospital. Failed Total Gneiss 110 1485 Granite 2 28 Loch raven schist 443 3733 2 3 4 Total Maﬁc 14 363 Total 5337 7048 5754 4405 22544 Marble 29 309 60 1403 LWBS 190 267 247 248 952 Prettyboy schist Admitted 1267 1538 669 964 4438 Other schists 46 933 17154 Serpentine 3 39 5243 4838 3193 Assume that the record is reviewed from a visit selected randomly from the table. a. What is the probability that the visit is LWBS or that the visit is from hospital 1? b. What is the probability that the visit is not LWBS and the hospital is 1 or 2? . WP Reserve Problem Chapter 2 Section 4 Problem 4 Consider the endothermic reactions in the table below. Final Temperature Conditions k Geological Formation Group 1 3880 Heat absorbed (cal) Below Target Above Target 266 K 21 40 271 K 44 27 274 K 56 36 Let A denote the event that a reaction’s ﬁnal temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Use the addition rules to calculate the following probabilities. a. P(A ∪ B) = b. P(A ∩ B′ ) = c. P(A′ ∪ B′ ) = . WP Reserve Problem Chapter 2 Section 4 Problem 5 Consider the well failure data in the table below. R-5 Wells Hospital Not admitted 8:50pm Page R-5 Let A denote the event that the geological formation of a well has more than 1000 wells, and let B denote the event that a well failed. Use the addition rules to calculate the following probabilities. a. P(A ∪ B) = b. P(A ∪ B′ ) = c. P(A′ ∪ B′ ) = . WP Reserve Problem Chapter 2 Section 4 Problem 6 Code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (ﬁve bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white) space appears between each bar. The original speciﬁcation (since revised) used exactly two wide bars and one wide space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally likely (none is held back as a delimiter). Determine the probability for each of the following: a. The ﬁrst bar is wide or the second bar is wide. b. Neither the ﬁrst nor the second bar is wide. c. The ﬁrst bar is wide or the second bar is not wide. d. The ﬁrst bar is wide or the ﬁrst space is wide. Exercises for Section 2.5 . WP Reserve Problem Chapter 2 Section 5 Problem 1 Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of orders in the previous month. The number of customers in each category are shown in the following table. Customers are to be selected randomly. Shipping Contract Express Standard Total No orders 22 17 39 One order 62 44 106 More than one order 40 23 63 k a. What is the conditional probability that the customer had one order last month given that the customer uses express shipping? b. What is the conditional probability that the customer uses express shipping given that the customer had at least one order last month? . WP Reserve Problem Chapter 2 Section 5 Problem 2 Consider the hospital emergency department visits shown in table below. People may leave without being seen by a physician, and those visits are denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may not be admitted for a stay in the hospital. k Trim Size: 8in x 10in STD-H8x10 R-6 k bend_RP.tex V1 - 11/08/2017 8:50pm Page R-6 Reserve Problems Hospital 1 2 3 4 Total Total LWBS Admitted Not admitted 5295 198 1277 3820 7172 271 1578 5323 5748 241 679 4828 4445 238 1014 3193 22660 948 4548 17164 Assume that the records are reviewed from two visits selected randomly without replacement from the table. a. What is the probability that the second visit selected is LWBS given that the ﬁrst visit selected is LWBS? b. What is the probability that both visits are LWBS from hospital 4? . WP Reserve Problem Chapter 2 Section 5 Problem 3 A committee will be formed with 4 managers and 3 engineers selected randomly without replacement from 11 managers and 19 engineers. What is the conditional probability that engineer Al is on the committee given that engineer Jane is on the committee? . WP Reserve Problem Chapter 2 Section 5 Problem 4 Consider the data on wafers classiﬁed by contamination and location in the table below. Number of Contamination Particles k Montgomery7e 0 1 2 3 4 5 or more Totals Center 0.28 0.17 0.10 0.06 0.04 0.07 0.72 Edge 0.12 0.03 0.05 0.04 0.01 0.03 0.28 a. Suppose a hacker selects a password at random. What is the probability that your password is selected? b. Suppose a hacker knows that your password is in event A and selects a password at random from this subset. What is the probability that your password is selected? c. Suppose a hacker knows that your password is in A and B and selects a password at random from this subset. What is the probability that your password is selected? . WP Reserve Problem Chapter 2 Section 5 Problem 7 If P(A|B) = 1, must A = B? Choose a correct Venn diagram to explain your answer. . WP Reserve Problem Chapter 2 Section 5 Problem 8 Consider the well failure data in the table below. Wells Geological Formation Group Failed Total Gneiss 130 1685 Granite 2 28 Loch raven schist 443 3733 Maﬁc 14 363 Totals Marble 32 309 0.40 0.20 0.15 0.10 0.05 0.10 1.00 Prettyboy schist 60 1403 Other schists 46 933 Serpentine 3 39 Assume that one wafer is selected at random from this set. Let A denote the event that a wafer contains four or more particles, and let B denote the event that a wafer is from the center of the sputtering tool. Determine the following probabilities. a. P(A) = b. P(A| B) = c. P(B) = d. P(B|A) = e. P(A ∩ B) = f. P(A ∪ B) = . WP Reserve Problem Chapter 2 Section 5 Problem 5 A batch of 370 samples of rejuvenated mitochondria contains 6 that are mutated (or defective). Two are selected from the batch, at random, without replacement. a. What is the probability that the second one selected is defective given that the ﬁrst one was defective? b. What is the probability that both are defective? c. What is the probability that both are acceptable? . WP Reserve Problem Chapter 2 Section 5 Problem 6 A computer system uses passwords that are exactly 5 characters and each character is one of the 26 letters (a–z) or 10 integers (0–9). You maintain a password for this computer system. Let A denote the subset of passwords that begin with a vowel (either a, e, i, o, or u) and let B denote the subset of passwords that end with some predetermined number (either 0, 1, 2, 3, 4, 6, 8 or 9). Let A denote the event that the geological formation of a well has more than 1000 wells, let B denote the event that a well failed, and let C denote the event that the geological formation of a well has fewer than 500 wells. a. What is the probability of a failure given there are more than 1,000 wells in a geological formation? b. What is the probability of a failure given there are fewer than 500 wells in a geological formation? . WP Reserve Problem Chapter 2 Section 5 Problem 9 Code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (ﬁve bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white) space appears between each bar. The original speciﬁcation (since revised) used exactly two wide bars and one wide space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally likely (none is held back as a delimiter). Determine the probability for each of the following. a. The second bar is wide given that the ﬁrst bar is wide. b. The third bar is wide given that the ﬁrst two bars are not wide. c. The ﬁrst bar is wide given that the last bar is wide. . WP Reserve Problem Chapter 2 Section 5 Problem 10 During a clinical trial the eﬀect of two treatments and a control for treatment of hepatitis C were considered. The following table provides the total patients in each group and the number that showed a complete (positive) response after 24 weeks of treatment. k k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems Complete Response Total Ribavirin plus interferon alfa 13 24 Interferon alfa 9 16 Untreated controls 0 20 Suppose that a patient is selected at random from this set. Let A denote the event that the patient is treated with ribavirin plus 8:50pm Page R-7 R-7 interferon alfa, and let B denote the event that the response is complete. Determine the following probabilities. a. P(B| A) = b. P(A|B) = c. P(A|B′ ) = d. P(A′ |B) = Exercises for Section 2.6 k . WP Reserve Problem Chapter 2 Section 6 Problem 1 Accidents on highways are one of the main causes of death or injury in developing countries and the weather conditions have an impact on the rates of death and injury. In foggy, rainy, and sunny conditions, 1/6, 1/10, and 1/25 of the accidents result in death, respectively. Sunny conditions occur 62% of the time, while rainy and foggy conditions each occur 19% of the time. What is the probability that an accident results in a death? . WP Reserve Problem Chapter 2 Section 6 Problem 2 A committee will be formed with 2 managers and 4 engineers selected randomly without replacement from 10 managers and 24 engineers. What is the probability that engineer Al and engineer Jane are on the committee? . WP Reserve Problem Chapter 2 Section 6 Problem 3 An article in the Transportation Research Part E Journal [“Arc Routing Problems to Restore Connectivity of a Road Network” (2016)] considered ways of re-establishing the connectivity of road networks after a natural disaster — earthquake. Estimates of the probabilities of a randomly chosen road being under light debris, moderate debris, and heavy debris conditions after diﬀerent disaster magnitudes are shown in the following table. Disaster magnitude is equally likely to be low, moderate or high. Disaster Magnitude Light Debris Moderate Debris Heavy Debris Total Low Moderate High Total 80 50 30 160 15 40 50 105 5 10 20 35 100 100 100 300 266 K 271 K 274 K Heat absorbed (cal) Below Target Above Target 19 44 56 40 11 36 k Hospital Total LWBS Admitted Not admitted 1 2 3 4 Total 5292 235 1277 3820 6991 230 1558 5163 5640 226 666 4728 4329 262 984 3103 22,252 953 4485 16,814 Suppose that three visits that resulted in LWBS are selected randomly (without replacement) for a follow-up interview. a. What is the probability that all three are selected from hospital 2? b. What is the probability that all three are from the same hospital? . WP Reserve Problem Chapter 2 Section 6 Problem 6 Consider the well failure data in the table below. a. What is the probability that a randomly selected road is under heavy debris after an earthquake? b. What is the probability that a randomly selected road is under light or moderate debris after an earthquake? . WP Reserve Problem Chapter 2 Section 6 Problem 4 Consider the endothermic reactions in the table below. Final Temperature Conditions Let A denote the event that a reaction’s ﬁnal temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Determine the following probabilities. a. P(A ∩ B) = b. P(A ∪ B) = c. P(A′ ∪ B′ ) = d. Use the total probability rule to determine P(A). . WP Reserve Problem Chapter 2 Section 6 Problem 5 The following table summarizes visits to emergency departments at four hospitals in Arizona. People may leave without being seen by a physician, and those visits are denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may not be admitted for a stay in the hospital. k Wells Geological Formation Group Failed Total Gneiss Granite Loch raven schist Maﬁc Marble Prettyboy schist Other schists Serpentine 170 2 383 14 29 60 46 3 2885 28 3733 363 309 1403 933 39 Trim Size: 8in x 10in STD-H8x10 R-8 bend_RP.tex V1 - 11/08/2017 8:50pm Page R-8 Reserve Problems Let A denote the event that the geological formation of a well has more than 1000 wells, and let B denote the event that a well failed. Determine the following probabilities. a. P(A ∩ B) = b. P(A ∪ B) = c. P(A′ ∪ B′ ) = d. Use the total probability rule to determine P(A). . WP Reserve Problem Chapter 2 Section 6 Problem 7 Consider the well failure data in the table below. Wells k Montgomery7e k Geological Formation Group Failed Total Gneiss 160 1685 Granite 2 28 Loch raven schist 453 3733 Maﬁc 14 363 Marble 29 309 Prettyboy schist 55 1403 Other schists 51 933 Serpentine 3 39 Suppose that two failed wells are selected randomly (without replacement) for a follow-up review. a. What is the probability that both are from the gneiss geological formation group? b. What is the probability that both are from the same geological formation group? . WP Reserve Problem Chapter 2 Section 6 Problem 8 Code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (ﬁve bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white) space appears between each bar. The original speciﬁcation (since revised) used exactly two wide bars and one wide space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally likely (none is held back as a delimiter). Determine the probability for each of the following: a. The code starts and ends with a wide bar. b. Two wide bars occur consecutively. c. Two consecutive wide bars occur at the start or end d. The middle bar is wide. . WP Reserve Problem Chapter 2 Section 6 Problem 9 A computer system uses passwords that contain exactly 5 characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let 𝛀 denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in 𝛀 are equally likely. Determine the following probabilities. a. P(A|B′ ) = b. P(A′ ∩ B′ ) = c. The probability that password contains exactly 2 integers given that it contains at least 1 integer. P = Exercises for Section 2.7 . WP Reserve Problem Chapter 2 Section 7 Problem 1 A cell phone user selects apps to download. Each of 10 apps is independently selected with probability 0.21. a. If each of the ﬁrst 9 apps are downloaded, what is the probability that the last app is downloaded? b. What is the probability that the cell phone user downloads at least 3 apps? c. What is the probability that app 1 or 2 is downloaded? . WP Reserve Problem Chapter 2 Section 7 Problem 2 An article in the Transportation Research Part E Journal [“Arc Routing Problems to Restore Connectivity of a Road Network” (2016)] considered ways of re-establishing the connectivity of road networks after a natural disaster — earthquake. Estimates of the probabilities of a randomly chosen road being under light debris, moderate debris, and heavy debris conditions after diﬀerent disaster magnitudes are shown in the following table. Disaster magnitude is equally likely to be low, moderate or high. Disaster Magnitude Low Light Debris Moderate Debris Heavy Debris Total 80 15 5 100 Moderate 50 40 10 100 High 30 50 20 100 Total 160 105 35 300 Let A and B denote the events that the earthquake magnitude is low and the road debris is heavy. Are these events independent? . WP Reserve Problem Chapter 2 Section 7 Problem 3 Suppose that a 9% discount is independently applied to Web orders for clothing with probability 0.150. a. What is the probability that the ﬁrst discount is applied to the ﬁfth order? b. What is the probability that at least one order in the next ﬁve receives the discount? k k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems . WP Reserve Problem Chapter 2 Section 7 Problem 4 Samples of emissions from three suppliers are classiﬁed for conformance to air-quality speciﬁcations. The results from 100 samples are summarized as follows: Supplier k No 1 21 9 2 25 5 3 30 10 Determine P(A). Determine P(B). Determine P(A ∩ B). Wells Let A denote the event that a sample is from supplier 1, and let B denote the event that a sample conforms to speciﬁcations. a. Are A and B independent events? b. Determine P(B|A). . WP Reserve Problem Chapter 2 Section 7 Problem 5 Six tissues are extracted from an ivy plant infested by spider mites. The plant in infested in 22% of its area. Each tissue is chosen from a randomly selected area on the ivy plant. a. What is the probability that there are exact four samples showing the signs of infestation and they are successive? b. What is the probability that there are exact three samples showing the signs of infestation and they are three out of four successive? . WP Reserve Problem Chapter 2 Section 7 Problem 6 In an acid-base titration, a base or acid is gradually added to the other until they have completely neutralized each other. Because acids and bases are usually colorless (as are the water and salt produced in the neutralization reaction), pH is measured to monitor the reaction. Suppose that the equivalence point is reached after approximately 100 mL of an NaOH solution has been added (enough to react with all the acetic acid present) but that replicates are equally likely to indicate from 95 to 104 mL, measured to the nearest mL. Assume that two technicians each conduct titrations independently. What is the probability that both technicians obtain equivalence at 101 mL? a. What is the probability that both technicians obtain equivalence between 98 and 102 mL (inclusive)? b. What is the probability that the average volume at equivalence from the technicians is 100 mL? . WP Reserve Problem Chapter 2 Section 7 Problem 7 Consider the endothermic reactions given below. Let A denote the event that a reaction’s ﬁnal temperature is 271 K or less. Let B denote the event that the heat absorbed is above target. Final Temperature Conditions R-9 Are A and B independent events? . WP Reserve Problem Chapter 2 Section 7 Problem 8 Consider the well failure data given below. Let A denote the event that the geological formation of a well has more than 1000 wells, and let B denote the event that a well failed. Conforms Yes 8:50pm Page R-9 Above Target 266K 11 41 271K 44 16 274K 56 36 Failed Total Gneiss 180 1985 Granite Loch raven schist 2 28 443 3733 Maﬁc 14 363 Marble 29 309 Prettyboy schist 60 1403 Other schists 46 933 Serpentine 3 39 Determine P(A). Determine P(B). Determine P(A ∩ B). Are A and B independent events? . WP Reserve Problem Chapter 2 Section 7 Problem 9 The code 39 is a common bar code system that consists of narrow and wide bars (black) separated by either wide or narrow spaces (white). Each character contains nine elements (ﬁve bars and four spaces). The code for a character starts and ends with a bar (either narrow or wide) and a (white) space appears between each bar. The original speciﬁcation (since revised) used exactly two wide bars and one wide space in each character. For example, if b and B denote narrow and wide (black) bars, respectively, and w and W denote narrow and wide (white) spaces, a valid character is bwBwBWbwb (the number 6). Suppose that all 40 codes are equally likely (none is held back as a delimiter). Let A denote the event that the ﬁrst bar is wide and B denote the event that the second bar is wide. Determine the following. Determine P(A). Determine P(B). Determine P(A ∩ B). Are A and B independent events? . WP Reserve Problem Chapter 2 Section 7 Problem 10 Consider the wafers categorized by location and contamination levels, so that the location is in the center or on the edge, and the contamination is low or high. Let the number of wafers with low contamination from the center and edge locations be denoted as nlc and nle , respectively. Similarly, let nhc and nhe denote the number of wafers with high contamination from the center and edge Heat absorbed (cal) Below Target Geological Formation Group k k Trim Size: 8in x 10in STD-H8x10 R-10 k Montgomery7e bend_RP.tex V1 - 11/08/2017 8:50pm Page R-10 Reserve Problems locations, respectively. Suppose that nlc = 12nhc and nle = 12nhe . That is, there are 12 times as many low contamination wafers as high ones from each location. Let A denote the event that contamination is low, and let B denote the event that the location is center. Are A and B independent? Does your conclusion change if the multiplier of 12 (between low and high contamination wafers) is changed from 12 to another positive integer? Exercises for Section 2.8 . WP Reserve Problem Chapter 2 Section 8 Problem 1 An article in the Transportation Research Part E Journal [“Arc Routing Problems to Restore Connectivity of a Road Network” (2016)] considered ways of re-establishing the connectivity of road networks after a natural disaster — earthquake. Estimates of the probabilities of a randomly chosen road being under light debris, moderate debris, and heavy debris conditions after diﬀerent disaster magnitudes are shown in the following table. Disaster magnitude is equally likely to be low, moderate or high. Disaster Magnitude k Light Debris Moderate Debris Heavy Debris Total Low 80 15 5 100 Moderate 50 40 10 100 High 30 50 20 100 Total 160 105 35 300 Given that a road had heavy debris after an earthquake, what is the conditional probability that the disaster magnitude was high? . WP Reserve Problem Chapter 2 Section 8 Problem 2 Accidents on highways are one of the main causes of death or injury in developing countries and the weather conditions have an impact on the rates of death and injury. In foggy, rainy, and sunny conditions, 1/4, 1/8, and 1/21 of the accidents result in death, respectively. Sunny conditions occur 60% of the time, while rainy and foggy conditions each occur 20% of the time. Given that an accident without deaths occurred, what is the conditional probability that it was foggy at the time? . WP Reserve Problem Chapter 2 Section 8 Problem 3 Consider the endothermic reactions given below. Final Temperature Conditions Use Bayes’ theorem to calculate the probability that a reaction’s ﬁnal temperature is 271 K or less given that the heat absorbed is above target. . WP Reserve Problem Chapter 2 Section 8 Problem 4 Consider the well failure data given below. Wells Geological Formation Group Failed Total Gneiss 170 1485 Granite 2 28 443 3733 Loch raven schist Maﬁc 14 363 Marble 29 309 Prettyboy schist 60 1403 Other schists 46 933 Serpentine 3 39 Use Bayes’ theorem to calculate the probability that a randomly selected well is in the gneiss group given that the well has failed. . WP Reserve Problem Chapter 2 Section 8 Problem 5 The probabilities of poor print quality given no printer problem, misaligned paper, high ink viscosity, or printer-head debris are 0, 0.3, 0.4, and 0.6, respectively. The probabilities of no printer problem, misaligned paper, high ink viscosity, or printer-head debris are 0.8, 0.02, 0.08, and 0.1, respectively. Determine the probability of high ink viscosity given poor print quality. Given poor print quality, what problem is most likely? Heat Absorbed (cal) Below Target Above Target 266K 13 39 271K 44 16 274K 56 36 k k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems 8:50pm Page R-11 R-11 Exercises for Section 2.9 . WP Reserve Problem Chapter 2 Section 9 Problem 1 Decide whether a discrete or continuous random variable is the best model for each of the following variables: a. The time a customer spends at a grocery store is _____. b. The number of arrivals to a grocery store on a given day is ______. c. The number of items each customer purchases at a grocery store is ________. d. The weekly revenue of a grocery store is _______. Supplemental Exercises for Chapter 2 k . WP Reserve Supplemental Exercises Chapter 2 Problem 1 Decide whether a discrete or continuous random variable is the best model for each of the following variables: a. The duration of a call at a mobile phone customer care service is _______. b. The number of busy representatives at a mobile phone customer care service at a time t is _______. c. The number of calls made to mobile phone customer care service on a given day is _____. . WP Reserve Supplemental Exercises Chapter 2 Problem 2 An article in the Transportation Research Part E Journal [“Arc Routing Problems to Restore Connectivity of a Road Network” (2016)] considered ways of re-establishing the connectivity of road networks after a natural disaster — earthquake. Estimates of the probabilities of a randomly chosen road being under light debris, moderate debris, and heavy debris conditions after diﬀerent disaster magnitudes are shown in the following table. Disaster magnitude is equally likely to be low, moderate or high. Disaster Magnitude Low Light Debris Moderate Debris Heavy Debris Total 80 15 5 100 Moderate 50 40 10 100 High 30 50 20 100 Total 160 105 35 300 What is the probability that a randomly selected road is under heavy debris given that the earthquake is either moderate or high? . WP Reserve Supplemental Exercises Chapter 2 Problem 3 Customers who are registered on a corporate Web site are summarized by the type of shipping contract they use and the number of orders in the previous month. The number of customers in each category are shown in the following table. Suppose that four customers are selected randomly, without replacement, for a survey. k Shipping Contract Express Standard Total No orders One order More than one order 30 73 40 18 50 28 48 123 68 a. What is the probability that three selected customers use express shipping and the other one uses standard shipping? b. What is the conditional probability that no selected customers ordered in the previous month given that all selected customers use express shipping? . WP Reserve Supplemental Exercises Chapter 2 Problem 4 A conceptual smartphone design uses a parachute technique to avoid severe damage to the screen and phone-case. Each of four nozzles located on the case might be either functional or defective after a year. The probability that a year-old nozzle is functional is 0.66 and assume that the nozzles fail independently. If at least two nozzles are functional, the phone will not be damaged in a drop. Otherwise, it will necessarily be damaged in a drop. a. What is the probability that a year-old phone is not damaged in a drop? b. What is the probability that all four nozzles of a year-old phone are functional? c. What is the conditional probability that all four nozzles are functional given that a year-old phone is not damaged in a drop? . WP Reserve Supplemental Exercises Chapter 2 Problem 5 A cell phone user selects apps to download. Each of 5 apps is independently selected with probability 0.2. Describe the sample space for the app downloads and determine the number of outcomes in the sample space. . WP Reserve Supplemental Exercises Chapter 2 Problem 6 A committee will be formed with 4 managers and 6 engineers selected randomly without replacement from 13 managers and 20 engineers. Consider the two events that engineer Jane and manager Mary are selected for the committee. Are these events independent? k Trim Size: 8in x 10in STD-H8x10 R-12 Hospital Total LWBS Admitted Not admitted 1 2 3 4 Total 5250 187 1239 3824 6877 255 1513 5109 5658 249 682 4727 4301 246 940 3115 22086 937 4374 16775 Determine the following probabilities. a. P(B|A) = b. P(A|B) = c. P(B′ |A) = d. P(A ∪ B′ ) = . WP Reserve Supplemental Exercises Chapter 2 Problem 10 A sample of three calculators is selected from a manufacturing line, and each calculator is classiﬁed as either defective or acceptable. Let A, B, and C denote the events that the ﬁrst, second, and third calculators, respectively, are defective. Sample space for this experiment could be described with a tree diagram. Let “d” denote a defective calculator and let “a” denote an acceptable calculator. calculator 1 d a calculator 2 d a d a a d calculator 3 d bend_RP.tex V1 - 11/08/2017 8:50pm Page R-12 Reserve Problems . WP Reserve Supplemental Exercises Chapter 2 Problem 7 Customers specify delivery dates for orders. Suppose that 14 customers each independently, randomly select delivery days over the next year (365 days). a. What is the probability that none select the same day? b. What is the probability that none select the same week? Assume exactly 52 weeks in a year. . WP Reserve Supplemental Exercises Chapter 2 Problem 8 You remove four fuses of 10, 20, 20, and 30 amperes each, but you do not mark the corresponding circuits. If you insert the fuses so that each sequence is equally likely, what is the probability that the appropriate amperage fuse is assigned to all the circuits? . WP Reserve Supplemental Exercises Chapter 2 Problem 9 Consider the hospital emergency room data from the table. Let A denote the event that a visit is to hospital 1 and let B denote the event that a patient is admitted to hospital 1. k Montgomery7e k a d a d a . WP Reserve Supplemental Exercises Chapter 2 Problem 11 Shafts are classiﬁed in terms of the machine tool that was used for manufacturing the shaft and conformance to surface ﬁnish and roundness. Tool 1 Surface Finish Conforms Roundness Conforms Yes No 197 4 4 2 Yes No Tool 2 Roundness Conforms Surface Finish Conforms Yes 145 10 Yes No No 4 4 a. If a shaft is selected at random, what is the probability that the shaft conforms to surface ﬁnish requirements or to roundness requirements or is from tool 1? b. If a shaft is selected at random, what is the probability that the shaft conforms to surface ﬁnish requirements or does not conform to roundness requirements or is from tool 2? c. If a shaft is selected at random, what is the probability that the shaft conforms to both surface ﬁnish and roundness requirements or the shaft is from tool 2? d. If a shaft is selected at random, what is the probability that the shaft conforms to surface ﬁnish requirements or the shaft is from tool 2? . WP Reserve Supplemental Exercises Chapter 2 Problem 12 The data from 200 machined parts are summarized as follows: Depth of Bore Edge Condition Coarse Moderate Smooth Above Target Below Target 15 24 48 10 21 82 a. What is the probability that a part selected has a moderate edge condition and a below-target bore depth? b. What is the probability that a part selected has a moderate edge condition or a below-target bore depth? c. What is the probability that a part selected does not have a moderate edge condition or does not have a below-target bore depth? . WP Reserve Supplemental Exercises Chapter 2 Problem 13 An e-mail message can travel through one of two server routes. The probability of transmission error in each of the servers and the proportion of messages that travel each route are shown in the following table. Assume that the servers are independent. S = {ddd, add, dda, ada, dad, aad, daa, aaa} a. Use the tree diagram to describe event A b. Use the tree diagram to describe event B c. Use the tree diagram to describe event A ∩ B d. Use the tree diagram to describe event B ∪ C Probability of Error Percentage of Messages Server 1 Server 2 Server 3 Server 4 Route 1 55 0.01 0.015 − − Route 2 45 − − 0.02 0.003 k k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems a. What is the probability that a message will arrive without error? b. If a message arrives in error, what is the probability it was sent through route 1? . WP Reserve Supplemental Exercises Chapter 2 Problem 14 A lot of 50 spacing washers contains 30 washers that are thicker than the target dimension. Washers are selected from the lot at random without replacement. a. What is the minimum number of washers that need to be selected so that the probability that all the washers are thinner than the target is less than 0.10? b. What is the minimum number of washers that need to be selected so that the probability that 1 or more washers are thicker than the target is at least 0.90? . WP Reserve Supplemental Exercises Chapter 2 Problem 15 Natural red hair consists of two genes. People with red hair have two dominant genes, two regressive genes, or one dominant and one regressive gene. A group of 1000 people was categorized as follows: 8:50pm Page R-13 R-13 Gene 2 Gene 1 Dominant Regressive Other Dominant 5 25 30 Regressive 7 63 35 Other 20 15 800 Let A denote the event that a person has a dominant red hair gene, and let B denote the event that a person has a regressive red hair gene. If a person is selected at random from this group, compute the following: a. P(A) = b. P(A ∩ B) = c. P(A ∪ B) = d. P(A′ ∩ B) = e. P(A|B) = f. Probability that the selected person has red hair. Chapter 3 Exercises Exercises for Section 3.1 k k . WP Reserve Problem Chapter 3 Section 1 Problem 1 Let us suppose that some article described that the primary charge transfer in a thunderstorm is due to collisions between soft hail particles and small crystals of ice at altitudes where the temperature generally ranges between −10 ∘ C and −20 ∘ C. Data are recorded after rounding the measured temperature in this range to the nearest one-tenth of a degree. Determine the range (possible values) of the random variable. . WP Reserve Problem Chapter 3 Section 1 Problem 2 Let us suppose that some article studied the probability of death due to burn injuries. The identiﬁed risk factors in this study are age greater than 60 years, burn injury in more than 40% of body-surface area, and presence of inhalation injury. It is estimated that the probability of death is 0.003, 0.03, 0.33, or 0.90, if the injured person has zero, one, two, or three risk factors, respectively. Suppose that three people are injured in a ﬁre and treated independently. Among these three people, two people have one risk factor and one person has three risk factors. Let the random variable X denote number of deaths in this ﬁre. Determine the probability mass function of X. . WP Reserve Problem Chapter 3 Section 1 Problem 3 Let us suppose that some article investigated the probability of corrosion of steel reinforcement in concrete structures. It is estimated that the probability of corrosion is 0.10 under speciﬁc values of half-cell potential and concrete resistivity. The risk of corrosion in ﬁve independent grids of a building with these values of half-cell potential and concrete resistivity is investigated now. Let the random variable X denote number of grids with corrosion in this building. Determine the probability mass function of X. k . WP Reserve Problem Chapter 3 Section 1 Problem 4 A Web site contains 100 interconnected pages. The random variable is the number of unique pages viewed by a visitor to the Web site. . WP Reserve Problem Chapter 3 Section 1 Problem 5 A disk drive manufacturer sells storage devices with capacities of one terabyte, 500 gigabytes, and 100 gigabytes with probabilities 0.5, 0.3, and 0.2, respectively. The revenues associated with the sales in that year are estimated to be $50 million, $25 million, and $10 million, respectively. Let X denote the revenue of storage devices during that year. Determine the probability mass function of X. . WP Reserve Problem Chapter 3 Section 1 Problem 6 The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows: Final Temperature Conditions Number of Reactions 266 K 48 271 K 60 274 K 92 Calculate the probability mass function of X. . WP Reserve Problem Chapter 3 Section 1 Problem 7 The following table shows the typical depth (rounded to the nearest foot) for nonfailed wells in geological formations in Baltimore County (The Journal of Data Science, 2009, Vol. 7, pp. 111–127). Trim Size: 8in x 10in STD-H8x10 R-14 Montgomery7e k bend_RP.tex V1 - 11/08/2017 8:50pm Page R-14 Reserve Problems Geological Formation Group Number of Nonfailed Wells Nonfailed Well Depth Gneiss Granite Loch Raven Schist Maﬁc Marble Prettyboy Schist Other schists Serpentine Total 1,515 26 3,290 349 280 1,343 887 36 7,726 255 218 317 231 267 255 267 217 2,027 Calculate the probability mass function of depth for nonfailed wells from the table. . WP Reserve Problem Chapter 3 Section 1 Problem 8 Consider the following circuit. The probability that each device functions correctly is p1 = 0.7 and p2 = 0.76. Assume that devices fail independently. p1 p2 Determine the probability mass function of X. Exercises for Section 3.2 k . WP Reserve Problem Chapter 3 Section 2 Problem 1 Determine the cumulative distribution function for the random variable in Exercise 3.1.12. . WP Reserve Problem Chapter 3 Section 2 Problem 2 Determine the cumulative distribution function for the random variable in Exercise 3.1.14. . WP Reserve Problem Chapter 3 Section 2 Problem 3 Determine the cumulative distribution function for the random variable X with the probability mass function f(x) = (4/5)(1/5). The range of X is {0, 1, 2, …}. . WP Reserve Problem Chapter 3 Section 2 Problem 4 The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows: . WP Reserve Problem Chapter 3 Section 2 Problem 6 The following table shows the typical depth (rounded to the nearest foot) for nonfailed wells in geological formations in Baltimore County (The Journal of Data Science, 2009, Vol. 7, pp. 111–127). Geological Formation Group Number of Nonfailed Wells Nonfailed Well Depth Gneiss 1,515 255 Granite 26 218 3,290 317 349 231 Loch Raven Schist Maﬁc 280 267 Prettyboy Schist 1,343 255 48 Other schists 887 267 271 K 60 Serpentine 274 K 92 Total Final Temperature Conditions Number of Reactions 266 K Detemine the cumulative distribution function for X. . WP Reserve Problem Chapter 3 Section 2 Problem 5 The distribution of the time until change (in days) of a Web site is approximated in the following table. Days until Changes Probability 1.5 0.05 3.0 0.25 4.5 0.35 5.0 0.20 7.0 0.15 Marble 36 217 7,726 2,027 Detemine the cumulative distribution function for X. Detemine the cumulative distribution function for X. k k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems 8:50pm Page R-15 R-15 Exercises for Section 3.3 . WP Reserve Problem Chapter 3 Section 3 Problem 1 Calculate the mean and variance for the random variable in Exercise 3.1.12. . WP Reserve Problem Chapter 3 Section 3 Problem 2 Calculate the mean and variance for the random variable in Exercise 3.1.14. . WP Reserve Problem Chapter 3 Section 3 Problem 3 The data from 200 endothermic reactions involving sodium bicarbonate are summarized as follows: Number of Contamination Particles Proportion of Wafers 0 0.30 1 0.20 2 0.15 Final Temperature Conditions Number of Reactions 3 0.10 266 K 271 K 274 K 48 60 92 4 0.05 5 or more 0.20 Calculate the mean and variance for X. . WP Reserve Problem Chapter 3 Section 3 Problem 4 The distribution of the time until change (in days) of a Web site is approximated in the following table. k . WP Reserve Problem Chapter 3 Section 3 Problem 6 A visual inspection of a location on wafers from a semiconductor manufacturing process resulted in the following table. Days until Changes Probability 1.5 3.0 4.5 5.0 7.0 0.05 0.25 0.35 0.20 0.15 Assume that wafers are independent with respect to contamination particles. Wafers are selected until one with ﬁve or more contamination particles occurs. Calculate the mean and variance for X. . WP Reserve Problem Chapter 3 Section 3 Problem 7 Consider the following circuit. The probability that each device functions correctly is p1 = 0.83 and p2 = 0.75. Assume that devices fail independently. p1 Calculate the mean for X. Calculate the mean and variance for X. . WP Reserve Problem Chapter 3 Section 3 Problem 5 The following table shows the typical depth (rounded to the nearest foot) for nonfailed wells in geological formations in Baltimore County (The Journal of Data Science, 2009, Vol. 7, pp. 111–127). Geological Formation Group Number of Nonfailed Wells Nonfailed Well Depth Gneiss Granite Loch Raven Schist Maﬁc Marble Prettyboy Schist Other schists Serpentine Total 1,515 26 3,290 349 280 1,343 887 36 7,726 255 218 317 231 267 255 267 217 2,027 Calculate the mean and variance for X. k p2 k Trim Size: 8in x 10in STD-H8x10 R-16 Montgomery7e k bend_RP.tex V1 - 11/08/2017 8:50pm Page R-16 Reserve Problems Exercises for Section 3.4 k . WP Reserve Problem Chapter 3 Section 4 Problem 1 The article “The Uniform Distribution as a First Practical Approach to New Product Inventory Management” [International Journal of Production Economics, 2008, 114(2)] proposed a uniform distribution to model the demand of a new product before observing the actual distribution. You will model demand forecasts with a discrete uniform distribution. According to your estimates, the minimum and the maximum levels of new product demand are 5 and 30 units per day, respectively. a. Determine the mean and variance of new product demand. b. Determine the probability mass function of new product demand. c. How do the mean and the variance of new product demand change, if you revise your estimate of maximum demand to be 20 instead of 30? . WP Reserve Problem Chapter 3 Section 4 Problem 2 The article “Statistical Method on Nonrandom Clustering with Application to Somatic Mutations in Cancer” [BMC Bioinformatics, 2011, 11(1)] developed a statistical method to discover mutations that lead to cancer by identifying nonrandom clusters of amino acid mutations in protein sequences in cells. In this method, N denotes the length of a protein sequence, and n denotes the total number of mutations in the protein. The position of a mutation is modeled with a discrete uniform distribution between 1 and N. Suppose that you are a biomedical engineer working on a protein sequence with two mutations and length N = 1000. Assume that the locations of the mutations are independent, and can be the same for two mutations. a. Determine the probability that the ﬁrst mutation occurs in the ﬁrst half of the protein sequence. b. Determine the probability that both of the mutations occur in the ﬁrst half of the protein sequence. c. Determine the probability that at least one of the mutations occurs in the ﬁrst half of the protein sequence. . WP Reserve Problem Chapter 3 Section 4 Problem 3 Show that for a discrete uniform random variable X, if each of the values in the range of X is multiplied by the constant c, the eﬀect is to multiply the mean of X by c and the variance of X by c2 . That is, show that E(cX) = cE(X) and V(cX) = c2 V(X). . WP Reserve Problem Chapter 3 Section 4 Problem 4 The number of pages in a PDF document you create has a discrete uniform distribution from ﬁve to nine pages (including the end points). What are the mean and standard deviation of the number of pages in the document? . WP Reserve Problem Chapter 3 Section 4 Problem 5 Suppose that nine-digit Social Security numbers are assigned at random. If you randomly select a number, what is the probability that it belongs to one of the 300 million people in the United States? . WP Reserve Problem Chapter 3 Section 4 Problem 6 The probability that data are entered incorrectly into a ﬁeld in a database is 0.005. A data entry form has 28 ﬁelds, and errors occur independently for each ﬁeld. The random variable X is the number of ﬁelds on the form with an error. Does X have a discrete uniform distribution? Why or why not? . WP Reserve Problem Chapter 3 Section 4 Problem 7 Consider the hospital data in the table. Suppose a patient is selected randomly from the collection in the table. Let X denote the hospital number of the selected patient (either 1, 2, 3, or 4). Does X have a discrete uniform distribution? Why or why not? Hospital Total LWBS Admitted Not admitted 1 2 3 4 Total 5350 195 1277 3878 5617 270 1558 3789 5102 246 666 4190 4472 242 984 3246 20541 953 4485 15103 Exercises for Section 3.5 . WP Reserve Problem Chapter 3 Section 5 Problem 1 Let us suppose that some article modeled the disease progression in sepsis (a systemic inﬂammatory response syndrome (SIRS) together with a documented infection). Both sepsis, severe sepsis and septic shock may be life-threatening. The researchers estimate the probability of sepsis to worsen to severe sepsis or septic shock after three days to be 0.10. Suppose that you are physician in an intensive care unit of a major hospital, and you diagnose four patients with sepsis. a. What is the probability that none of the patients with sepsis gets worse in the next three days? b. What is the probability that all of the patients with sepsis get worse in the next three days? c. What is the probability that at most two patients with sepsis get worse in the next three days? . WP Reserve Problem Chapter 3 Section 5 Problem 2 An article in Fire Technology, 2014 (50.3) studied the eﬀectiveness of sprinklers in ﬁre control by the number of sprinklers that activate correctly. The researchers estimate the probability of a sprinkler to activate correctly to be 0.7. Suppose that you are an inspector hired to write a safety report for a large ballroom with 10 sprinklers. Assume the sprinklers activate correctly or not independently. a. What is the probability that all of the sprinklers will operate correctly in a ﬁre? k k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems b. What is the probability that at least 7 of the sprinklers will operate correctly in a ﬁre? c. What is the minimum number of sprinklers needed so that the probability that at least one operates correctly is at least 0.99? . WP Reserve Problem Chapter 3 Section 5 Problem 3 Sketch the probability mass function of a binomial distribution with n = 10 and p = 0.01 and comment on the shape of the distribution. a. What value of X is most likely? b. What value of X is least likely? . WP Reserve Problem Chapter 3 Section 5 Problem 4 A particularly long traﬃc light on your morning commute is green 20% of the time that you approach it. Assume that each morning represents an independent trial. a. Over 5 mornings, what is the probability that the light is green on exactly one day? b. Over 20 mornings, what is the probability that the light is green on exactly four days? c. Over 20 mornings, what is the probability that the light is green on more than four days? . WP Reserve Problem Chapter 3 Section 5 Problem 5 Consider the patient data in table. Suppose that four patients are randomly selected with replacement from the total for hospital 1. Hospital k 1 2 3 4 Total Total 5892 6991 5640 4329 22852 LWBS 195 270 246 242 953 Admitted 1277 1558 666 984 4485 Not admitted 4420 5163 4728 3103 17414 Determine the following probabilities: a. Exactly one is LWBS. b. Two or more are LWBS. c. At least one is LWBS. . WP Reserve Problem Chapter 3 Section 5 Problem 6 Assume that a Web site changes its content according to the distribution in the table below. 8:50pm Page R-17 R-17 b. What is the probability that the change is made in less than 4 days in 2 or fewer of the 10 updates? c. What is the probability that at least one change is made in less than 4 days? d. What is the expected number of the 10 updates that occur in less than 4 days? . WP Reserve Problem Chapter 3 Section 5 Problem 7 Consider the endothermic reactions in the table below. Final Temperature Conditions Number of Reactions 226 K 48 271 K 60 274 K 92 A total of 20 independent reactions are to be conducted. a. What is the probability that exactly 12 reactions result in a ﬁnal temperature less than 272 K? b. What is the probability that at least 19 reactions result in a ﬁnal temperature less than 272 K? c. What is the probability that at least 18 reactions result in a ﬁnal temperature less than 272 K? d. What is the expected number of reactions that result in a ﬁnal temperature of less than 272 K? . WP Reserve Problem Chapter 3 Section 5 Problem 8 Consider the circuit. k pw a b pw Suppose that the probability of a device functioning correctly is pw = 0.75 and assume that devices fail independently. What is the probability mass function of the number of device failures? . WP Reserve Problem Chapter 3 Section 5 Problem 9 Consider the patient data in the table below. Suppose that patients are randomly selected with replacement from the total for hospital 4. Days untill changes Probability 1.5 0.05 1 2 3 4 Total 3.0 0.25 Total 5299 5965 5178 5047 21489 4.5 0.35 LWBS 195 270 246 242 953 5.0 0.20 Admitted 1277 1558 666 984 4485 7.0 0.15 Not admitted 3827 4137 4266 3821 16051 Hospital Assume that 10 changes are made independently. a. What is the probability that the change is made in less than 4 days in 7 of the 10 updates? k What is the smallest sample size needed so that the probability is at least 90% that at least one patient is LWBS? Trim Size: 8in x 10in STD-H8x10 R-18 Montgomery7e k bend_RP.tex V1 - 11/08/2017 8:50pm Page R-18 Reserve Problems Exercises for Section 3.6 k . WP Reserve Problem Chapter 3 Section 6 Problem 1 A robot wrestling tournament with eight participants is taking place. The defending champion is expected to win a match with the probability of 0.85 regardless of the opponent, and matches outcomes are assumed to be independent. a. The single elimination tournament requires three consecutive match wins to win the tournament. What is the probability that the defending champion wins the tournament? b. The defending champion won the tournament again and now accepts open challenges. What is the expected number of matches until this robot is defeated by a challenger? c. After the ﬁrst defeat, the robot’s joints are replaced to more ﬂexible ones, increasing the winning probability to 0.94. What is the probability that this robot’s ﬁrst loss is the ﬁfth challenge? . WP Reserve Problem Chapter 3 Section 6 Problem 2 A research team has developed a face recognition device to match photos in a database. From laboratory tests, the recognition accuracy is 92% and trials are assumed to be independent. a. If the research team continues to run laboratory tests, what is the mean number of trials until failure? b. What is the probability that the ﬁrst failure occurs on the tenth trial? c. To improve the recognition algorithm, a chief engineer decides to collect 10 failures. How many trials are expected to be needed? . WP Reserve Problem Chapter 3 Section 6 Problem 3 A fault-tolerant system that processes transactions for a ﬁnancial services ﬁrm uses three separate computers. If the operating computer fails, one of the two spares can be immediately switched online. After the second computer fails, the last computer can be immediately switched online. Assume that the probability of a failure during any transaction is 10-9 and that the transactions can be considered to be independent events. a. What is the mean number of transactions before all computers have failed? b. What is the variance of the number of transactions before all computers have failed? . WP Reserve Problem Chapter 3 Section 6 Problem 4 Show that the probability density function of a negative binomial random variable equals the probability density function of a geometric random variable when r = 1. Show that the formulas for the mean and variance of a negative binomial random variable equal the corresponding results for a geometric random variable when r = 1. . WP Reserve Problem Chapter 3 Section 6 Problem 5 Consider the endothermic reactions in the table below. Assume that independent reactions are conducted. Final Temperature Conditions Number of Reactions 226 K 271 K 274 K 48 60 92 a. What is the probability that the ﬁrst reaction to result in a ﬁnal temperature less than 272 K is the 12th reaction? b. What is the mean reaction number of the ﬁrst ﬁnal temperature is less than 272 K? c. What is the probability that the ﬁrst reaction to result in a ﬁnal temperature less than 272 K occurs within 3 or fewer reactions? d. What is the mean number of reactions until two reactions result in ﬁnal temperatures less than 272 K? . WP Reserve Problem Chapter 3 Section 6 Problem 6 The following table summarizes visits to emergency departments at four hospitals. People may leave without being seen by a physician, and those visits are denoted as LWBS. The remaining visits are serviced at the emergency department, and the visitor may or may not be admitted for a stay in the hospital. Hospital Total LWBS Admitted Not admitted 1 2 3 4 Total 5463 195 1277 3991 5642 270 1558 3814 5235 246 666 4323 4029 242 984 2803 20369 953 4485 14931 Suppose that patients are randomly selected with replacement, from the total for hospital 4. Determine the following: a. Probability that the ﬁrst patient admitted is the ﬁrst one selected. b. Probability that four or fewer patients are selected to admit two. c. Expected number of patients selected to admit 20. Exercises for Section 3.7 . WP Reserve Problem Chapter 3 Section 7 Problem 1 A technology company is forming a task force of six members to deal with urgent quality issues. The positions will be ﬁlled by randomly chosen qualiﬁed applicants. The qualiﬁed applicants consist of ﬁve managers and ten engineers. a. What is the probability that the chosen applicants are either all managers or all engineers? b. What is the probability that number of managers is the same as the number of engineers on the task force? k k Trim Size: 8in x 10in STD-H8x10 Montgomery7e k bend_RP.tex V1 - 11/08/2017 Reserve Problems k c. What is the expected number of engineers chosen? d. What is the probability that at least one manager is chosen for the task force? . WP Reserve Problem Chapter 3 Section 7 Problem 2 The MIT-BIT arrhythmia database contains 48 heart signal recordings with 22 and 26 from female and male patients, respectively. For a classiﬁcation task, an analyst randomly selects 36 records for predictive model training and keeps the other 12 records for testing the model performance. a. What is the expected number of female patient recordings selected for training? b. What is the probability that at least three male patient recordings are selected for training? c. What is the probability that the same number of male and female patients are used for testing the model performance? . WP Reserve Problem Chapter 3 Section 7 Problem 3 Calculate the ﬁnite population corrections for the following situations. a. Suppose that X has a hypergeometric distribution with N = 140, n = 4, K = 50. If ﬁnite population correction factor is small a binomial distribution can eﬀectively approximate the hypergeometric distribution. Calculate the following probabilities, assuming that X has a binomial distribution. b. Printed circuit cards are placed in a functional test after being populated with semiconductor chips. A lot contains 150 cards, and 20 are selected without replacement for functional testing. Use the binomial approximation to the hypergeometric distribution to approximate the following probabilities. If 23 cards are defective, what is the probability that at least 1 defective card is in the sample? If 7 cards are defective, what is the probability that at least 1 defective card appears in the sample? . WP Reserve Problem Chapter 3 Section 7 Problem 4 Consider the visits that result in leave without being seen (LWBS) at an emergency department in the table below. Assume that four visits that result in LWBS are to be randomly selected (without replacement) for a follow-up interview. Hospital # Total 1 2 3 4 Total 5292 6991 5640 4329 22,252 LWBS 195 270 246 242 953 Admitted 1277 1558 666 984 4485 Not admitted 3820 5163 4728 3103 16,814 a. What is the probability that all selected visits are from hospital 4? b. What is the probability that no selected visits are from hospital 4? c. What is the probability that all selected visits are from the same hospital? k 8:50pm Page R-19 R-19 . WP Reserve Problem Chapter 3 Section 7 Problem 5 Consider the nonfailed wells in the table below. Assume that four wells are selected randomly (without replacement) for inspection. Geological Formation Group Number of Nonfailed Wells Nonfailed Well Depth 1,515 26 3,290 349 280 1,343 887 36 7,726 255 218 317 231 267 255 267 217 2,027 Gneiss Granite Loch Raven Schist Maﬁc Marble Prettyboy Schist Other Schists Serpentine Total a. What is the probability that exactly two are selected from the Loch Raven Schist? b. What is the probability that one or more is selected from the Loch Raven Schist? c. What is the expected number selected from the Loch Raven Schist? . WP Reserve Problem Chapter 3 Section 7 Problem 6 Consider the semiconductor wafer data in the table below. Location in Sputtering Tool Contamination Low Center Edge Total 514 68 582 358 High 112 246 Total 626 314 Suppose that 10 wafers are selected randomly (without replacement) for an electrical test. Determine the following: a. Probability that exactly 4 wafers have high contamination. b. Probability that at least 1 is from the center of the sputtering tool and has high contamination. c. Probability that exactly 3 have high contamination or are from the edge of the sputtering tool. d. Instead of 10 wafers, what is the minimum number of wafers that need to be selected so that the probability that at least 1 wafer has high contamination is greater than or equal to 0.9? k Trim Size: 8in x 10in STD-H8x10 R-20 k Montgomery7e bend_RP.tex V1 - 11/08/2017 8:50pm Page R-20 Reserve Problems Exercises for Section 3.8 k . WP Reserve Problem Chapter 3 Section 8 Problem 1 An engineer is participating in a research project on the title patterns of junk emails. The number of junk emails which arrive in an individual’s account every hour follows a Poisson distribution with a mean of 1.2. a. What is the expected number of junk emails that an individual receives in an eight-hour day? b. What is the probability that an individual receives more than two junk emails for the next three hours? c. What is the probability that an individual receives no junk email for two hours? . WP Reserve Problem Chapter 3 Section 8 Problem 2 The arrivals of patients at a walk-in clinic between 1:00 and 2:00 PM follows a Poisson process with the mean of 10.5. a. What is the probability that 15 patients arrive at the clinic between 1:00 and 2:00 PM? b. What is the probability that no patients arrive between 1:00 and 1:10 pm? c. Suppose that 20 patients arrive between 1:00 and 1:30 PM. What is the probability that three more patients arrive between 1:30 and 2:00 PM? d. Suppose that the arrivals from 12:00 to 1:00 PM follows a Poisson process with a mean of 4.7 Is the number of arrivals between 12:00 and 2:00 PM a Poisson random variable? Why or why not? . WP Reserve Problem Chapter 3 Section 8 Problem 3 The number of ﬂaws in bolts of cloth in textile manufacturing is assumed to be Poisson distributed with a mean of 0.1 ﬂaw per square meter. a. What is the probability that there are two ﬂaws in one square meter of cloth? b. What is the probability that there is one ﬂaw in 10 square meters of cloth? c. What is the probability that there are no ﬂaws in 20 square meters of cloth? d. What is the probability that there are at least two ﬂaws in 10 square meters of cloth? . WP Reserve Problem Chapter 3 Section 8 Problem 4 The number of content changes to a Web site follows a Poisson distribution with a mean of 0.25 per day. a. What is the probability of two or more changes in a day? b. What is the probability of no content changes in ﬁve days? c. What is the probability of two or fewer changes in ﬁve days? . WP Reserve Problem Chapter 3 Section 8 Problem 5 Orders arrive at a Web site according to a Poisson process with a mean of 12 per hour. Determine the following: a. Probability of no orders in ﬁve minutes. b. Probability of 3 or more orders in ﬁve minutes. c. Length of a time interval such that the probability of no orders in an interval of this length is 0.001. Supplemental Exercises for Chapter 3 . WP Reserve Supplemental Exercises Chapter 3 Problem 1 For each of the following exercises, determine the range (possible values) of the random variable. a. A store manager is interested in the customer traﬃc to a Web site from midnight to 8:00 AM. She counts the number unique visitors to the site in that time period. b. An article in the Journal of Materials Science: Materials in Medicine 2010 21(4) studies the eﬀect of the surface ﬁnish of the metal stem on the longevity of the implants in total hip replacement surgeries. Random damage events were examined with emitted acoustic signals under the application of a direct shear force. The total number of random damage events is 2624 ±235 at the yield stage. A material science researcher models the total number of damage events with the grit-blasted surface ﬁnish at the yield stage as a random variable assuming that its range is within the observed values in this article. . WP Reserve Supplemental Exercises Chapter 3 Problem 2 Typically, a job shop is a manufacturing system in which a variety of custom products are produced in small batches. In the study “Analysis of Reactive Scheduling Problems in a Job Shop Environment” [European Journal of Operational Research, 2000, 126(3)], the researchers tested the performance of a variety of job scheduling policies in a job shop system with machine breakdowns. In these tests, it is assumed that the number of operations required to complete the production of a job has a discrete uniform distribution between 5 and 15. In addition, the processing time of each operation has discrete uniform distribution between 20 and 80 minutes. a. Determine the mean and variance of the number of operations required to complete the production of a job. b. Determine the mean and variance of the processing time of an operation. c. Can one conclude that the mean of the total time to complete all required operations of a job is 500 minutes? Why or why not? . WP Reserve Supplemental Exercises Chapter 3 Problem 3 The proton exchange membrane (PEM) fuel cells produce electricity with hydrogen fuel and oxygen of the air. An article in the Journal of Power Sources, 2008, 178(1) studied a two-phase ﬂow (a ﬂow of gas and liquid) model applied to the gas diﬀusion layer of PEM fuel cells. Five diﬀerent values of channel width are k k Trim Size: 8in x 10in STD-H8x10 k Montgomery7e bend_RP.tex V1 - 11/08/2017 Reserve Problems k considered with equal probability. These values are 0.3, 0.4, 0.5, 0.6 and 0.7 mm. Let the random variable X denote the channel width. Determine the probability mass function of X. . WP Reserve Supplemental Exercises Chapter 3 Problem 4 An article in Soil Dynamics and Earthquake Engineering 2008, 28.10 evaluated the vulnerability of buildings in Barcelona, Spain to earthquakes. For instance, they estimate that the probability of a high-rise, unreinforced masonry building located in zone II of Barcelona has a moderate or severe damage is approximately 0.46. Assume that a dozen high-rise, unreinforced masonry buildings located in zone II of Barcelona respond independently. Determine the probabilities of the following outcomes from an earthquake: a. All 12 buildings have moderate or severe damage b. At least ten of these buildings have moderate or severe damage c. Fewer than two or more than ten of these buildings have moderate or severe damage . WP Reserve Supplemental Exercises Chapter 3 Problem 5 An article in The BMJ 2014, 4.9 studied the cost-eﬀectiveness of nalmefene (a drug used mainly in the treatment of alcohol dependence) with psychosocial support in decreasing alcohol consumption. Suppose that the probability of a patient to remain in abstinence (successfully respond) after a year of treatment with nalmefene and psychosocial support is 0.20. The same probability is estimated to be 0.08 if the patient is treated with only psychosocial support. Suppose that you are a physician who have ten new alcohol dependent patients, and you plan to provide psychosocial support to all of them and also provide treatment with nalmefene to ﬁve. Assume that patient respond independently. a. Is the number of patients who successfully respond a binomial random variable? Why or why not? b. What is the probability that at least two patients will remain in abstinence after a year of treatment with nalmefene and psychosocial support? c. What is the probability that exactly two of the ten patients successfully respond? (Hint: consider each group of patients separately.) . WP Reserve Supplemental Exercises Chapter 3 Problem 6 Customers of a hardware shop make a payment either in cash or with credit/debit card with probabilities 0.3 and 0.7, respectively. Assume these probabilities apply to all customers independently. a. If 20 customers pay at the hardware shop, what is the probability that exactly 5 customers pay in cash? b. What is the probability that the second customer to pay cash is the ﬁfth customer to pay at the shop? c. Assume that customers arrive according to a Poisson process with a mean of 10.5 per hour. What is the probability that there are no cash transactions in a period of 30 minutes? . WP Reserve Supplemental Exercises Chapter 3 Problem 7 Equipment maintenance on a manufacturing line either replaces or cleans a ﬁlter. In 85% of the maintenance events the ﬁlter is cleaned. Assume that ﬁlter events occur independently. a. What is the probability that the ﬁlter is ﬁrst replaced after it is cleaned twice? b. What is the mean and variance of the number of maintenance events until the ﬁlter is replaced? k 8:50pm Page R-21 R-21 c. Suppose that the number of days until maintenance events follows a geometric distribution with a mean of 5 days. How would you determine the mean number of days until the ﬁlter is replaced and what assumptions would you make? . WP Reserve Supplemental Exercises Chapter 3 Problem 8 A food company is conducting research on customers’ taste. In each round of a blind taste experiment four black teas and one herbal tea are presented to participants. Suppose that the participants randomly guess, independently in each round. a. If a participant picks the herbal tea in each of three rounds, a box of tea is won. What is the probability that a participant wins a box of tea? b. What is the expected number of participants until a second box of tea is won? . WP Reserve Supplemental Exercises Chapter 3 Problem 9 The number of contamination particles (that exceed a speciﬁed size) in display panels for televisions follows the Poisson distribution with the mean of 0.0001 per square inch. A quality engineer is inspecting a television panel that is 55 by 35 inches. a. What is the probability that there is particle in the panel? b. If there are at most two particles, the panel is sent to the assembly shop. What is the probability that a panel is sent to the assembly shop? c. What is the probability that at least 9 of the next 10 panels are sent to the assembly shop? Assume that panels are independent with respect to contamination. . WP Reserve Supplemental Exercises Chapter 3 Problem 10 Batches that consist of 50 coil springs from a production process are checked for conformance to customer requirements. The mean number of nonconforming coil springs in a batch is ﬁve. Assume that the number of nonconforming springs in a batch, denoted as X, is a binomial random variable. a. What are n and p? b. What is P(X _ 2)? c. What is P(X _ 49)? . WP Reserve Supplemental Exercises Chapter 3 Problem 11 The probability is 0.6 that a calibration of a transducer in an electronic instrument conforms to speciﬁcations for the measurement system. Assume that the calibration attempts are independent. What is the probability that at most three calibration attempts are required to meet the speciﬁcations for the measurement system? . WP Reserve Supplemental Exercises Chapter 3 Problem 12 The probability that your call to a service line is answered in less than 30 seconds is 0.85. Assume that your calls are independent. a. If you call 15 times, what is the probability that exactly 9 of your calls are answered within 30 seconds? b. If you call 20 times, what is the probability that at least 16 calls are answered in less than 30 seconds? c. If you call 25 times, what is the mean number of calls that are answered in less than 30 seconds? d. What is the probability that you must call four times to obtain the ﬁrst answer in less than 30 seconds? e. What is the mean number of calls until you are answered in less than 30 seconds? . WP Reserve Supplemental Exercises Chapter 3 Problem 13 The probability that your call to a service line is answered in less than 30 seconds is 0.6. Assume that your calls are independent. k Trim Size: 8in x 10in STD-H8x10 k Montgomery7e bend_RP.tex V1 - 11/08/2017 8:50pm Page R-22 Reserve Problems R-22 a. What is the probability that you must call six times in order for two of your calls to be answered in less than 30 seconds? b. What is the mean number of calls to obtain two answers in less than 30 seconds? . WP Reserve Supplemental Exercises Chapter 3 Problem 14 Consider the circuit shown in the ﬁgure above where the probability that each device functions correctly is provided. Assume that all devices fail independently. What is the probability of two or fewer failed devices? 0.9 . WP Reserve Supplemental Exercises Chapter 3 Problem 15 Determine the constant so that the following function is a probability mass function: f(x) = cx for x = 1, 2, 3, 4. . WP Reserve Supplemental Exercises Chapter 3 Problem 16 Suppose that 50 sites on a patient might contain lesions. A biopsy selects 8 sites randomly (without replacement). a. What is the minimum number of sites with lesions so that the probability of at least one selected site contains lesions is greater than or equal to 0.95? b. Rework the above section if the probability of that event is greater than or equal to 0.99. 0.95 a 0.9 0.99 b 0.95 0.9 Chapter 4 Exercises Exercises for Section 4.1 k . WP Reserve Problem Chapter 4 Section 1 Problem 1 A parabolic satellite dish reﬂects signals to the dish’s focal point. An antenna designer analyzed signals transmitted to a satellite ( dish and) obtained the probability density function f (x) = 1 c 1 − x2 for 0 < x < 2, where X is the distance (in meters) 16 from the centroid of the dish surface to a reﬂection point at which a signal arrives. Determine the following: a. Value or c that makes f(x) a valid probability density function b. P(X ≤ 0.4) = c. P(0.1 < X < 0.4) = . WP Reserve Problem Chapter 4 Section 1 Problem 2 The talk time (in hours) on a cell phone in a month is approxix − 10 for 10 < mated by the probability density function f (x) = 5h x − 25 1 for 20 ≤ x ≤ 25. Determine x 0). Use the cumulative distribution function to 1000 Exercises for Section 4.3 k . WP Reserve Problem Chapter 4 Section 3 Problem 1 A parabolic satellite dish reﬂects signals to the dish’s focal point. An antenna designer analyzed signals transmitted to a satellite ( dish and) obtained the probability density function f (x) = 1 c 1 − x2 for 0 < x < 3, where x is the distance (in meters) 16 from the centroid of the dish surface to a reﬂection point at which a signal arrives. Calculate the mean and variance. E(X) = V(X) = . WP Reserve Problem Chapter 4 Section 3 Problem 2 The talk time (in hours) on a cell phone in a month is approxix − 10 for 10 < mated by the probability density function f (x) = 5h x − 25 1 for 20 ≤ x ≤ 25. Determine x 2.0) = e. P(0 < Z < 0.7) = . WP Reserve Problem Chapter 4 Section 5 Problem 4 Assume that X is normally distributed with a mean of 7 and a standard deviation of 4. Determine the value for x that solves each of the following equations. a. P(X > x) = 0.5
b. P(X > x) = 0.95
c. P(x < X < 9) = 0.2 d. P(3 < X < x) = 0.95 e. P(−x < X − 7 < x) = 0.99 . WP...

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