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.

Don't use plagiarized sources. Get Your Custom Essay on

George Washington University Statistics Worksheet

Just from $13/Page

Why Work with Us

Top Quality and Well-Researched Papers

We always make sure that writers follow all your instructions precisely. You can choose your academic level: high school, college/university or professional, and we will assign a writer who has a respective degree.

Professional and Experienced Academic Writers

We have a team of professional writers with experience in academic and business writing. Many are native speakers and able to perform any task for which you need help.

Free Unlimited Revisions

If you think we missed something, send your order for a free revision. You have 10 days to submit the order for review after you have received the final document. You can do this yourself after logging into your personal account or by contacting our support.

Prompt Delivery and 100% Money-Back-Guarantee

All papers are always delivered on time. In case we need more time to master your paper, we may contact you regarding the deadline extension. In case you cannot provide us with more time, a 100% refund is guaranteed.

Original & Confidential

We use several writing tools checks to ensure that all documents you receive are free from plagiarism. Our editors carefully review all quotations in the text. We also promise maximum confidentiality in all of our services.

24/7 Customer Support

Our support agents are available 24 hours a day 7 days a week and committed to providing you with the best customer experience. Get in touch whenever you need any assistance.

Try it now!

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

No need to work on your paper at night. Sleep tight, we will cover your back. We offer all kinds of writing services.

Essays

No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.

Admissions

Admission Essays & Business Writing Help

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.

Reviews

Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.

Reviews

Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied with the service offered.