1. For events A, B, we are given: P( B ) = 0.3 ,
What is P( Ac ∪ B ) ?
P( B – A ) = 0.12 ,
A and B are independent events.
2. If you roll two fair dice repeatedly, what is the probability that you will get a sum of 4 before you
get a sum of 5 ?
( Hint: What does the 7 on the denominator tell you? )
3. You have 3 dice. For each die: Three sides are painted yellow, two sides are painted red, and one
side is painted blue. If the three dice are rolled, what is the probability that they all land with the
same color facing up?
4. Three boys and 3 girls are seated randomly in a row. What is the probability that the boys sit
together and the girls sit together?
5. A toolbox contains 12 bolts and 10 nuts. If a handyman grabs 8 items randomly, what is the
probability that he gets four bolts and four nuts? (The answer is rounded to three decimal places).
6. In a certain town, 70% of the homes have a fireplace, 40% have a hot tub, and 80% have a fireplace or
a hot tub. Given that a home has a hot tub, what is the probability that it has a fireplace?
7. There are 3 coins in a box – a nickel, a dime, and a quarter.
These are not fair coins. Each coin has different probabilities for Heads and Tails.
For the nickel:
P( Head ) = 1/3.
For the dime:
P( Head ) = 2/3
For the quarter: P( Head ) = 1.
A coin is chosen at random from the box and tossed once. Given that it showed Heads, what is the
probability that the quarter was chosen?
8. Let X be a discrete random variable with the pmf: 𝑝𝑝(𝑥𝑥) =
What is P( X > 25 ) ?
9. Let X be a discrete random variable with the pmf:
What is the standard deviation of 3X ?
x = 2, 3, 4, …..
+ 0.2 ,
x = 0, 10, 30.
(The answer is rounded to 2 decimal places).
10. You roll a fair die 12 times. What is the probability that (exactly) two rolls will show a ‘6’ ?
(The answer is rounded to 3 decimal places).
Hint: what random variable is this?
11. The number of errors per minute in a signal processor is a Poisson random variable with mean 6.
What is the probability that there is at most one error in a period of 30 seconds?
(a) e –6
(b) 3 e –6
(c) e –3
(d) 3 e –3
(e) 4 e –3
12. Let X be a continuous random variable with the probability density function:
f (x) = c ( 1 − x2 ) , −1 < x < 1. (a) 0.1 (b) 0.2 What is the expected value of X2 ? (c) 0.3 (d) 0.4 13. Let X be a continuous random variable with the probability density function: 𝑥𝑥 f (x) = , 0 < x < c. What is the median of X ? 64 (a) 5.6 (b) 6.4 (c) 7.2 (d) 8.0 (e) 0.5 (e) 8.8 14. Buses from Denver to Colorado Springs leave at 8:20, 8:40 and 9:00. Buses from Denver to Fort Collins leave at 8:25 and 8:50. These are the only buses that leave the bus station between 8:00 and 9:00. You will arrive at the bus station randomly at a uniformly distributed time between 8:00 and 9:00, and you will get on the next bus that leaves. What is the probability that you will go to Fort Collins? (a) 0.20 (b) 0.25 (c) 0.30 (d) 0.35 (e) 0.40 15. The diameter of ostrich eggs is a normally distributed with µ = 165 and σ = 20 (mm). What is the probability that the diameter of an ostrich egg will exceed 161 mm? (a) 0.3286 (b) 0.3907 (c) 0.4512 (d) 0.5140 (e) 0.5793 16. The amount of rainfall in a certain place is a normal random variable with a mean of 28, and a standard deviation of 5. The probability that the amount of rain will exceed the capacity of the local reservoir is 0.10. What is the capacity of the reservoir? (a) 34.4 (b) 37.5 (c) 40.6 (d) 44.2 (e) 48.3 17. A data set is presented by the frequency histogram below. What percent of the data values are between 12 and 17? (including 12 and 17) Histogram of the Data Set 9 8 7 Frequency 6 5 4 3 2 1 0 1 2 3 4 (a) 0.30 5 6 7 8 (b) 0.34 9 10 11 x (c) 0.38 12 13 14 15 (d) 0.42 16 17 18 19 20 (e) 0.46 18. Below is a frequency table with only the cumulative frequencies shown. What is the sample mean? (a) 5.1 Distinct Values Cumulative Frequencies 4 3 5 10 6 20 7 24 8 24 9 30 (b) 5.5 (c) 5.9 (d) 6.3 (e) 6.7 19. What is the sample median of the following data set? 28, 36, 10, 65, 13, 47, 23, 21, 32, 54, 23, 40, 18 (a) 23 (b) 25.5 (c) 28 (d) 30 (e) 32 20. A researcher compared test scores between two school districts. A sample in district A gave a sample mean of 72, and a sample standard deviation of 10. A sample in district B gave a sample mean of 114, and a sample standard deviation of 20. Scoring 64 in district A is comparable to what score in district B? (a) 98 (b) 92 (c) 86 (d) 80 (e) 74 21. A data set yielded a sample mean of 18, and a sample standard deviation of 4. Using z-score, how small does an observation in this data set have to be in order to be considered an outlier? (a) Smaller than 6 (b) Smaller than 8 (d) Smaller than 12 (e) Smaller than 14 (c) Smaller than 10 22. The daily milk production of a new breed of cows has a standard deviation of 5 gallons. Researchers plan to sample 100 cows in order to estimate the mean amount. What is the probability that the sampling error will not exceed 0.6 gallon? (a) 0.0956 (b) 0.2743 (c) 0.5478 (d) 0.7699 (e) 0.8845 23. Suppose X is a random variable with σ = 40. A sample of size 16 gave a sample mean of 162. What is the 99% confidence interval for the true mean of X? (a) ( 120.85 , 203.15 ) (b) ( 124.58 , 199.42 ) (c) ( 128.75 , 195.25 ) (d) ( 132.53 , 191.47 ) (e) ( 136.25 , 187.75 ) 24. A bird watcher studies a species of birds whose weight is known to have a standard deviation of 8 grams. She wants to obtain a 90% confidence interval for the mean weight of these birds, with the length of the interval not exceeding 4. How many birds does she need to sample? (a) 37 (b) 43 (c) 44 (d) 62 (e) 63 25. A sample of size 25 was taken, and the sample standard deviation came out to be 6.25. If we compute the 95% confidence interval for the true mean, what will be our margin of error? (a) ± 2.58 (b) ± 2.38 (c) ± 2.18 (d) ± 2.78 (e) ± 2.98 26. A forester wants to test if the mean height of pine trees exceeds 55 feet. A sample of 64 trees yielded a sample mean of 59.2525 feet, and a sample standard deviation of 14 feet. What is the value of the test statistic? (a) 2.03 (b) 2.13 (c) 2.23 (d) 2.33 (e) 2.43 27. We want to test whether or not the mean of X equals 150 at significance level 0.08. What is the critical value? (a) 1.405 (b) 1.552 (c) 1.645 (d) 1.751 (e) 1.96 28. Suppose we want to test H0: µ = 60 Vs. Ha: µ ≠ 60, at significance level α = 0.08. A sample of size n = 100 yielded a sample mean of 57.1, and a sample standard deviation of 20. Which of the following is the right conclusion? (a) P-value = 0.0735, therefore we reject H0 (b) P-value = 0.0885, therefore we fail to reject H0 (c) P-value = 0.147, therefore we reject H0 (d) P-value = 0.0735, therefore we fail to reject H0 (e) P-value = 0.147, therefore we fail to reject H0 29. A clothing manufacturer wanted to compare the mean thermal insulation of cotton fabrics and triacetate fabrics. A sample of 36 cotton fabrics gave � 1 = 12.3 and s1 = 3. 𝑿𝑿 and s2 = 2. � 2 = 8.6 A sample of 48 triacetate fabrics gave 𝑿𝑿 What is the 93% confidence interval for the difference between the mean insulation of cotton and triacetate fabrics? (a) ( 2.938 , 4.462 ) (b) ( 2.416 , 4.984 ) (d) ( 3.188 , 4.212 ) (e) ( 2.654 , 4.746 ) (c) ( 2.152 , 5.248 ) 30. We want to test if there is a difference between the mean lifetime (in hours) of two brands of batteries. We sampled 50 batteries of brand 1, and then 50 batteries of brand 2. The following results were obtained: � 1 = 118.6 , 𝑿𝑿 � 2 = 111.2 , 𝑿𝑿 s1 = 20 , s2 = 20. We want to test H0: µ1 = µ2 Vs. Ha: µ1 ≠ µ2 , at significance level 0.01. What is the value of the test statistic? (a) 1.65 (b) 1.85 (c) 2.05 (d) 2.25 (e) 2.45 31. A dentist wanted to estimate the proportion of people who floss every day. He conducted a survey that included 300 randomly selected people, and found that 132 of them floss every day. Give the 92% confidence interval for the true proportion. (a) ( 0.39 , 0.49 ) (b) ( 0.37 , 0.51 ) (d) ( 0.41 , 0.47 ) (e) ( 0.43 , 0.45 ) (c) ( 0.35 , 0.53 )
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