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STAT240 Section: Name: Solution
Read the following directions carefully. DO NOT turn to the next page until the exam has started.
Write your name and section number at the top right of this page:
Class Section Number
Bret 8:50 001
Sahifa 1:20 003
Miranda 9:55 004
Sahifa 3:30 005
Sahifa 8:50 006
Cameron 1:20 007
As you complete the exam, write your initials at the top right of each other page.
If you need more room, there is a blank page at the end of the exam, or we can give you some scratch paper.
Some multiple choice questions are “Select ONE” while others are “Select ALL that apply”. Pay attention to the question type and only mark one option if it says “Select ONE”. Fill in the circles completely.
1. The number of credits that a UW-Madison course is worth follows this distribution, with some numbers removed (Not real data).
x 1 2 3 4
P(X = x) ? ? 0.4 0.4
(a) Suppose P(X = 1) and P(X = 2) are equal. What are P(X = 1) and P(X = 2)?
(b) Write out a mathematical expression for E[X], no need to evaluate or simplify.
2. The PACER test is a fitness test used to assess aerobic fitness. It involves running back and forth between two lines which are twenty meters apart. An audio series of ”beeps” is played, and you must make it to the opposite line before the next beep sounds. If you do so, you have completed ”a lap”. If you fail to do so, the test is over. The beeps get closer together over time.
Consider one person running the PACER test, and let X be the number of laps they successfully complete (e.g. make it to the other side before the beep). Does X follow a binomial distribution? Briefly comment on each of the individual assumptions and why it is or is not met.
3. A spinner at a carnival will either land on “Lose” or “Win” when spun. Assume spins are independent and each one has the same probability of landing on “Win”.
(a) (3 points) The carnival claims the spinner has a win probability of p = 0.4. Assuming this is correct, which lines of R code below can be used to find the probability of winning exactly twice in 3 spins? Select ALL that apply.
0.6*0.6dbinom(2, 3, 0.4)pbinom(2, 3, 0.4)
0.6*0.6*0.4dbinom(2, 3, 0.6)pbinom(2, 3, 0.6)
3*0.6*0.4*0.4dbinom(1, 3, 0.6)pbinom(1, 3, 0.4)
(b) (3 points) You suspect that the true win probability p is actually less than 0.4. Write a null and alternative hypothesis to test the value of p.
(c) (2 points) You decide to complete the test with significance level α = 0.05. You spin the spinner
10 times and win 2 times, giving a p-value of 167. What is the most accurate conclusion of the test? Select ONE.
Because our p-value is greater than α, we do not have evidence of p < 0.4.
Because our p-value is greater than α, we have evidence of p < 0.4.
We have evidence that the true p = 0.4.
We have evidence that the true p = 0.2.
We do not have enough information to determine the distribution.
5. We are interested in the true percentage, p, of UW-Madison students who know that Bucky Badger’s full first name is Buckingham. We ask 100 students, and construct a 90% confidence interval for p as (0.50, 0.70).
(a) Which of the following correctly calculates the critical value for a 90% CI? Select ONE.
qnorm(0.9)qnorm(0.975) qnorm(0.95)dbinom(2, 3, 0.4)
(b) Which of the following statements are TRUE about this interval? Select ALL that apply.
A 95% confidence interval would have a larger margin of error than this one.
A 95% confidence interval would have the same margin of error as this one.
If we had asked 500 students AND decreased confidence level to 80%, the resulting interval would be narrower than this one.
If we had asked 500 students AND decreased confidence level to 80%, we do not know if the resulting interval would be narrower or wider than this one.
(c) Which of the following conclusions can be made from this confidence interval? Select ONE.
At least half of all UW-Madison students know this fact. p is below 0.8.
60% of the students sampled knew this fact.
We are 95% confident that 60% of all UW students know this fact.
Write your name and section number at the top right of this page:
Class Section Number
Bret 8:50 001
Sahifa 1:20 003
Miranda 9:55 004
Sahifa 3:30 005
Sahifa 8:50 006
Cameron 1:20 007
As you complete the exam, write your initials at the top right of each other page.
If you need more room, there is a blank page at the end of the exam, or we can give you some scratch paper.
Some multiple choice questions are “Select ONE” while others are “Select ALL that apply”. Pay attention to the question type and only mark one option if it says “Select ONE”. Fill in the circles completely.
1. The number of credits that a UW-Madison course is worth follows this distribution, with some numbers removed (Not real data).
x 1 2 3 4
P(X = x) ? ? 0.4 0.4
(a) Suppose P(X = 1) and P(X = 2) are equal. What are P(X = 1) and P(X = 2)?
(b) Write out a mathematical expression for E[X], no need to evaluate or simplify.
2. The PACER test is a fitness test used to assess aerobic fitness. It involves running back and forth between two lines which are twenty meters apart. An audio series of ”beeps” is played, and you must make it to the opposite line before the next beep sounds. If you do so, you have completed ”a lap”. If you fail to do so, the test is over. The beeps get closer together over time.
Consider one person running the PACER test, and let X be the number of laps they successfully complete (e.g. make it to the other side before the beep). Does X follow a binomial distribution? Briefly comment on each of the individual assumptions and why it is or is not met.
3. A spinner at a carnival will either land on “Lose” or “Win” when spun. Assume spins are independent and each one has the same probability of landing on “Win”.
(a) (3 points) The carnival claims the spinner has a win probability of p = 0.4. Assuming this is correct, which lines of R code below can be used to find the probability of winning exactly twice in 3 spins? Select ALL that apply.
0.6*0.6dbinom(2, 3, 0.4)pbinom(2, 3, 0.4)
0.6*0.6*0.4dbinom(2, 3, 0.6)pbinom(2, 3, 0.6)
3*0.6*0.4*0.4dbinom(1, 3, 0.6)pbinom(1, 3, 0.4)
(b) (3 points) You suspect that the true win probability p is actually less than 0.4. Write a null and alternative hypothesis to test the value of p.
(c) (2 points) You decide to complete the test with significance level α = 0.05. You spin the spinner
10 times and win 2 times, giving a p-value of 167. What is the most accurate conclusion of the test? Select ONE.
Because our p-value is greater than α, we do not have evidence of p < 0.4.
Because our p-value is greater than α, we have evidence of p < 0.4.
We have evidence that the true p = 0.4.
We have evidence that the true p = 0.2.
We do not have enough information to determine the distribution.
5. We are interested in the true percentage, p, of UW-Madison students who know that Bucky Badger’s full first name is Buckingham. We ask 100 students, and construct a 90% confidence interval for p as (0.50, 0.70).
(a) Which of the following correctly calculates the critical value for a 90% CI? Select ONE.
qnorm(0.9)qnorm(0.975) qnorm(0.95)dbinom(2, 3, 0.4)
(b) Which of the following statements are TRUE about this interval? Select ALL that apply.
A 95% confidence interval would have a larger margin of error than this one.
A 95% confidence interval would have the same margin of error as this one.
If we had asked 500 students AND decreased confidence level to 80%, the resulting interval would be narrower than this one.
If we had asked 500 students AND decreased confidence level to 80%, we do not know if the resulting interval would be narrower or wider than this one.
(c) Which of the following conclusions can be made from this confidence interval? Select ONE.
At least half of all UW-Madison students know this fact. p is below 0.8.
60% of the students sampled knew this fact.
We are 95% confident that 60% of all UW students know this fact.
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