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[answered] Midterm #2 - Spring 2016 - PSTAT 109 May 20, 2016 Test Form

Answers for 1-4 with work. Please

1. (4 points) Suppose we have a random sample X1;X2;X3;X4 that follow a Normal dis-

tribution. Let be the standard deviation and be the mean. What is the distribution

of the random variable

X

????

=

p

n?

(a) t distribution with 4 degrees of freedom.

(b) t distribution with 3 degrees of freedom.

(c) Normal distribution with mean and standard deviation p

4

.

(d) The Standard Normal distribution.

(e) The Continuous Uniform distribution.

2. (4 points) Let random variable K have a Binomial distribution with n = 240 and =

0:20. Which is the correct application of the continuity correction for probability that

we observe at least 45 successes. (Recall: K is the number of successes out of n trials.)

(a) P(K 44:5)

(b) P(K 45)

(c) P(K 45:5)

(d) P(K 45)

(e) P(K 44:5)

3. (4 points) Let X have Normal distribution with E[X] = 16 and Var[X] = 25. Which

Excel function would compute P(X 24)?

(a) NORMDIST(24, 16, 25, TRUE)

(b) NORMDIST(24, 16, 5, TRUE)

(c) NORMDIST(24, 16, 25, FALSE)

(d) NORMDIST(16, 24, 5, FALSE)

(e) NORMDIST(24, 16, 5, FALSE)

Page 3

4. (4 points) Do I have an unfair six-sided die? I roll a die two times and if I roll two

dierent face values I will conclude that my die is fair. Here are my hypotheses:

H0 : the die is not fair vs. Ha : the die is fair. Which statement is an example of Type

I error?

(a) In reality the die is fair, and I roll a 1 and a 6.

(b) In reality the die is not fair, and I roll a 4 and a 5.

(c) In reality the die is fair, and I roll a 2 and a 2.

(d) In reality the die is not fair, and I roll a 3 and a 3.

(e) I forgot. Doh!

Midterm #2 ? Spring 2016 ? PSTAT 109

May 20, 2016

Test Form B Name (Please Print) :

Seat: Has the person sitting to YOUR RIGHT given you a valid photo ID that matches the one

written on the top of their cover sheet? Their formula sheet does NOT have any additional

notes? If the seat to your right is empty, check the ID of the next person to the right. Yes No There is no person to my right Has the person sitting to YOUR LEFT given you a valid photo ID that matches the one

written the top of their cover sheet? Their formula sheet does NOT have any additional

notes? If the seat to your left is empty, check the ID of the next person to the left. Yes No There is no person to my left Name (Please Print): Signature: Midterm #2 ? Spring 2016 ? PSTAT 109

Test Form B

May 20, 2016 Name (Please Print): UCSB Perm Number: Seat: Academic Integrity: On midterm exams and the final exam, the standard rules apply

as outlined in ?Academic Integrity at UCSB: A Student?s Guide.? On those exams, you are

cheating if you copy from another student, consult notes or a textbook during the exam,

have another student take an exam for you, and communicate with another student about

answers to the exam. Electronic devices such as cell phones, computers, bluetooth devices

etc. are not permitted. Any student who cheats during an exam will receive an F for the

course, and the incident will be reported to the Office of Judicial Affairs for further sanctions.

Possible sanctions include suspension and dismissal. You can anonymously report cheating

incidents. The email is nava@pstat.ucsb.edu. This exam is 40 minutes in duration. You must remain in your seat for the

entire exam period ? you may not leave early for any reason.

Each multiple choice question is worth 4 points. In the long answer section

possible points for each question (or sub-question) are listed at the beginning of

the question.

In the long answer section you must completely prove all answers ? unless

otherwise stated, undefended answers receive zero points. Answers that are not

completely proven and defended will not receive full credit. Page 2 This exam is 40 minutes in duration. You must remain in your seat for the

entire exam period ? you may not leave early for any reason.

Each multiple choice question is worth 4 points. In the long answer section

possible points for each question (or sub-question) are listed at the beginning of

the question.

In the long answer section you must completely prove all answers ? unless

otherwise stated, undefended answers receive zero points. Answers that are not

completely proven and defended will not receive full credit. 1 Multiple Choice: Only 1 correct answer per problem. Record answer on scantron. Test Form B.

1. (4 points) Suppose we have a random sample X1 , X2 , X3 , X4 that follow a Normal distribution. Let ? be the standard deviation and ? be the mean. What is the distribution

?

X??

? ?

of the random variable ?/

n

(a) t distribution with 4 degrees of freedom.

(b) t distribution with 3 degrees of freedom.

(c) Normal distribution with mean ? and standard deviation ?? .

4 (d) The Standard Normal distribution.

(e) The Continuous Uniform distribution.

2. (4 points) Let random variable K have a Binomial distribution with n = 240 and ? =

0.20. Which is the correct application of the continuity correction for probability that

we observe at least 45 successes. (Recall: K is the number of successes out of n trials.)

(a) P(K ? 44.5)

(b) P(K ? 45)

(c) P(K ? 45.5)

(d) P(K ? 45)

(e) P(K ? 44.5)

3. (4 points) Let X have Normal distribution with E[X] = 16 and Var[X] = 25. Which

Excel function would compute P(X ? 24)?

(a) NORMDIST(24, 16, 25, TRUE)

(b) NORMDIST(24, 16, 5, TRUE)

(c) NORMDIST(24, 16, 25, FALSE)

(d) NORMDIST(16, 24, 5, FALSE)

(e) NORMDIST(24, 16, 5, FALSE) Page 3 4. (4 points) Do I have an unfair six-sided die? I roll a die two times and if I roll two

different face values I will conclude that my die is fair. Here are my hypotheses:

H0 : the die is not fair vs. Ha : the die is fair. Which statement is an example of Type

I error?

(a) In reality the die is fair, and I roll a 1 and a 6.

(b) In reality the die is not fair, and I roll a 4 and a 5.

(c) In reality the die is fair, and I roll a 2 and a 2.

(d) In reality the die is not fair, and I roll a 3 and a 3.

(e) I forgot. Doh!

5. (4 points) Suppose Qianyu constructs a 92% confidence interval for the population

mean. Using the same data, Bernard constructs a 90% confidence interval for the population mean. In comparison to Qianyu?s confidence interval, Bernard?s interval is:

(a) Exactly the same.

(b) Wider than Qianyu?s confidence interval.

(c) Narrower than Qianyu?s confidence interval.

(d) Shifted to the right of Qianyu?s confidence interval.

(e) Shifted to the left of Qianyu?s confidence interval. Page 4 Name: Perm Number: Section Day/Time: 6. (40 points) The UCSB Arbor espresso vending machine is set to fill a cup with my

favorite espresso beverage. The amount of each fill varies. Suppose the amount of fills

are independent and follow a Normal distribution with average of 7.1 ounces and a

standard deviation of 0.4 ounces. (All solutions must be written on this page.)

(a) (8 points) Compute the 5th percentile of the espresso machine?s fill amounts. (b) (8 points) I order four espressos. What is the probability that the total amount

will be exactly 28.4 ounces? (c) (12 points) I order four espressos. What is the probability that the average fill

amount will be below 7 ounces? (d) (12 points) I order four espressos. What is the probability that all four are filled

below the 6.85 ounces? Page 5 7. (40 points) After taking the Democratic vote in West Virginia and Nebraska from former Secretary of State Hillary Clinton in the primaries, Sanders is looking forward to

sweeping Kentucky and Oregon as part of his predicted ?May winning streak.? While

this may sound ridiculous to some, American voters are still feeling the Bern and well,

so is Sanders himself. We sampled 353 registered Democrats in Kentucky and Oregon

and find 155 support Clinton over Sanders. From our survey, we want to test whether

Clinton will obtain less than half of the votes. (All responses MUST be written on this

side of sheet only.)

(a) (10 points) What is the population and parameter of interest in this study? (b) (10 points) Write the null and alternative hypotheses. (c) (10 points) Compute the appropriate test statistic. (d) (10 points) Compute the p-value and state your conclusion using ? = 0.02. Page 6

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