George Washington University Statistics Worksheet

ExamThe exam covers material in Chapters 16 through 21. You may use the text, R, your notes, your
prior homework, and other personal work for this take-home exam. HOWEVER, make sure
your work is indeed your work.
1. (5 points) Federal law under Title 49 of the United States Code, Chapter 301, Motor Vehicle
Safety Standard took effect on January 1, 1968 and required all vehicles (except buses) to be
fitted with seat belts in all designated seating positions. While most states have laws requiring
seat belt use today, some people still do not “buckle up.” Let’s assume that 90 % of drivers do
“buckle up.” If drivers are randomly stopped to check seat belt usage, answer the following
questions and show your work.
a) (1 point) How many drivers do they expect to stop before finding a driver whose seatbelt
is not buckled?
b) (1 point) What is the probability that the second unbelted driver is in the ninth car stopped?
c) (1 point) What is the probability that of the first 10 drivers, 8 or more are wearing their
seatbelts?
d) (1 point) If they stop 30 cars during the first hour, find the mean and standard deviation of
the number of drivers not expected to be wearing seatbelts?
e) (1 point) If they stop 120 cars during this safety check, what is the probability they find at
least 12 drivers not wearing seatbelts?
2. (5 points) Unwanted calls (including illegal and spoofed robocalls) are the FCC’s top
consumer complaint. The United States is the 8th most spammed country in the world, and
the annoying calls are on the rise according to a new report. Suppose that a spammer is
testing a scheme to get people to buy something over the phone and getting the “customer” to
provide credit card information. He wants to test his scheme in the following way. He has
hacked another company’s customer list containing all 200,000 of its customers’ phone
numbers. He randomly calls 1,000 of these customers, and he is able to get 123 of the called
customers to reveal their credit card information.
a) (1 point) Create a 90% confidence interval for the true proportion p for all 200,000
customers on his list who might reveal credit card information if in fact he decides to call
all 200,000 of them. Be sure to check all necessary assumptions and conditions.
b) (1 point) Explain what your interval means by explaining what “90% confidence” means
in this context.
c) (1 point) The scammer only wants to call all 200,000 people on the customer list if he
thinks he will be able to convince at least 5% of them to reveal their credit card
information. What does you confidence interval imply about this?
d) (1 point) In the interval you constructed in a), the probability that the true population
proportion p is actually in your specified interval is .90. True or False (and if false,
why)?
e) (1 point) Generally speaking, for two confidence intervals with the same level of
confidence and with random samples from the same population, the interval with the
larger sample size has a better chance of containing the population parameter being
estimated. True of False (and if false, why)?
3. (5 points) Suppose that in manufacturing a very sensitive electronic component, a company
and its customers have tolerated a 2% defective rate. Recently, however, several customers
have been complaining that there seem to be more defectives than in the past. Given that the
company has made recent modifications to its manufacturing process, it is wondering if in fact
the defective rate has increased from 2%. For quality assurance purposes, you decide to
randomly select 1,000 of these electronic components before they are shipped to customers.
Of the 1,000 components, you find 25 that are defective. Assume that the company produces
a very large number of these components on any given day.
a) (1 point) Set up an appropriate hypothesis to test whether or not the defect rate has
increased.
b) (1 point) Before proceeding to test your hypothesis, check that all assumptions and
conditions are satisfied for such a test.
c) (2 points) Conduct the test using a .05 level of significance (alpha) and state your decision
about whether or not you believe that the defect rate has increased.
d) (1 point) What would be the minimum number of defectives in a random sample of 1,000
would you need to find in order to statistically decide that the defect rate exceeds .02 (again,
assuming a .05 level of significance).
Answer:
Given x=25
n=1000
alpha = 0.05
(a) setting up hypothesis
Null hypothesis : defective rate of components is 2%
Alternate hypothesis: defective rate of components has increased from 2%
(b) Since company produces a very large number of these components on any given day, we can
assume
, where N is total number of components produced.
np = 1000 * 0.02 = 200 >10
np(1-p)=1000*0.02*(1-0.02) = 19.6 > 10
Hence we can assume shape of sampling distribution of sample proportion is approximately normal.
(c) sampling proportion
Since it is right tailed test
z-critical value when alpha is 0.05 is = 1.64
p-value (area to the right of z-test = 1.13) = 0.1294
Since
or p-value is greater than level of significance ( alpha = 0.05) , we failed to
reject the null hypothesis.
This means there is not enough evidence to support the claim that defective components rate has
been increased from 2%
(d)
when alpha is 0.05, z-critical = 1.64
z-test statistics should be greater than this value for rejecting the null hypothesis and concluding that
defective rate has been increased from 2%
Rounding to nearest next integer, minimum number of defectives should be 28 in order to
statistically decide that the defective rate has been increased.
4. (5 points) Along with interest rates, life expectancy is a component in pricing financial
annuities. Suppose that you know that last year average life expectancy was 77 years for your
annuity holders. Now you want to know if your clients this year have a longer life expectancy,
on average, so you randomly sample n=20 of your recently deceased annuity holders to see
actual age at death. Using a 5% level of significance, test whether or not the new data shows
evidence of your annuity holders now live longer than 77 years, on average. The data below
are the sample data (in years of life):
(78,75,83,81,81,77,78,79,79,81,76,79,77,76,79,81,73,74,78,79)
a) (2 points) Does this sample indicate that life expectancy has increased? Test an appropriate
hypothesis and state your conclusion (use a 5% level of significance). Be sure to check the
necessary assumptions and conditions before conducting your test.
b) (2 points) Construct A 90% confidence interval for the true average age of death for the
population of your annuity holders. Explain why your confidence interval agrees or not
statistically with your hypothesis testing decision in part a).
c) (1 point) Suppose that you want to construct 90% confidence interval that has a margin of
error of one half of a year. What size sample would you need at a minimum?
5. (5 points) If you want to know how important spam filters are to your online experience, try
turning them off for a day. You’ll quickly see why these tools we tend to take for granted are
so essential. Generally speaking, a filtering solution applied to your email system uses a set
of protocols to determine which incoming messages are spam and which are not. What the
filters checks on can vary, but often they all do basically the same thing: scan header
information for evidence of malice, look up senders on blacklists of known spammers, and
filter content for patterns that point to junk mail.
Suppose that a particular spam filter uses a points-based system in which various aspects of an
email trigger an accumulation of points – with 100 points being the maximum and strongly
indicating spam. So, more points for a particular email becomes stronger evidence that it is
spam. After accumulating a sufficient number of points, the spam filter classifies the email as
spam and it does not reach your inbox.
This process is similar to hypothesis testing in the following way for each email it reviews:
H0: The email is a real message (not spam)
HA: The email is spam
Using the above hypothesis setting context, answer the following questions using
language/terms we have covered related to hypothesis testing:
a) (1 point) When the filter allows spam to slip through into your inbox, which kind of error is
that? Explain in terms of the hypotheses above.
b) (1 point) Which kind of error is it when a real (i.e., non-spam) email gets classified as spam
and does not get to your inbox? Explain in terms of the hypotheses above.
c) (1 point) Suppose that this particular spam filter classifies spam as any email getting 50
points or higher. However, you reset the filter to use 60 points or higher before classifying
it as spam. Is that analogous to choosing a higher or lower alpha level for a hypothesis test.
Explain in terms of the hypotheses above.
d) (1 point) What impact does this change in the spam cutoff value have on the chance of each
type of error in hypothesis testing? Explain.
e) (1 point) What does “power” mean in this context of the spam filter, and how is it related to
one of the two types of errors? Explain in terms of the hypotheses above.