Economics 120A Final Examination

a.    You have 3 hours to finish your exam. Write your name, ID number and your assigned seat on the upper left corner of this page. If you are not seated in your correct seat assignment, your score will be penalized by 10 points.

b.    WRITE YOUR STUDENT ID NUMBER IN THE UPPER CORNER OF ALL THE ODD PAGES IN THIS TEST. YOUR TEST HAS 18 PAGES TOTAL.

c.    Make sure you find the formula sheets, as well as a standard normal cumulative distribution table, in the last pages of your exam. You may (and should) detach those last two pages of your exam.

d.    All answers must be written on the attached sheets, in the spaces provided.

e.    There are two parts to this exam – multiple-choice questions (Part I) and short-answer questions (Part II). You do not need to justify your answers for the multiple-choice questions. Show ALL your work for the Part II questions.

f.    KEEP YOUR EYES ON YOUR OWN PAPER AND KEEP YOUR PAPER OUT OF VIEW OF YOUR NEIGHBORS.

g.    You are not allowed to use your cell phone during the exam. Please, turn off your cell phone and put it away. If you are found using a cellphone during the final examination, you will be considered in violation of academic integrity, and you will be reported to the Academic Integrity Office.

h.    If you need scratch blank paper, or if you have any questions about the exam, please ask.

PART I: Multiple-Choice Questions (1.5 points each, 27 points total).

You do NOT need to justify your answers for the multiple-choice questions. Please, fill your answers here. USE CAPITAL LETTERS.

1. 2.

1)    Consider the following two statements:

I.   Suppose that random samples are drawn from a population. The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean is approximately Normal for any n, regardless of the distribution of the population.

II.   The sampling distribution of the sample mean will have the same standard deviation as the original population from which the samples were drawn.

A)   Both statements are true.

B)   Both statements are false.

C)   Only statement I is true.

D)   Only statement II is true.

2)    Three Statistics classes all took the same test. Here are the histograms of the scores for each class:

Consider the following statements:

I. Class 3 has the highest median score.

II. Class 1 has the smallest standard deviation.

A)    Only statement I is true.

B)    Only statement II is true.

C)    Both statements are true

D)    Both statements are false.

9. 10. 11. 12. 13.

9)   Which of these statements is false?

A)    The expected value of the sample average is equal to the population mean.

B)    As the sample size gets larger, the variance of the sample average increases.

C)    If the parent population is normally distributed, then the sampling distribution of the sample average is also normally distributed, regardless of the sample size.

D)    If the parent population is not normal, the sampling distribution of the sample average approaches normality as the sample size increases.

10) Which of the following statements is not correct?

A)   Two events A and B are mutually exclusive if the two events cannot occur together.

B)   If event A does not occur, then its complement (A) will also not occur.

C)   If events A and B occur at the same time, then A and B intersect.

D)   A union of events (A or B or C) occurs when at least one of the events occurs.

11) A professor of linguistics refutes the claim that the average time a student spends studying for the midterm exam is 3 hours. She thinks they spend more time than that. Which hypotheses are used to test the claim?

A)    H0: μ > 3 vs. H1: μ = 3

B)    H0: μ = 3 vs. H1: μ ≠ 3

C)    H0: μ = 3 vs. H1: μ > 3

D)    H0: X = 3 vs. H1: X > 3

12)  The life of a new type of light bulb is uniformly distributed between 1,200 and 1,600 hours. There is a 70% probability that a randomly-selected light bulb will last at most how long?

A)    1320 hours

B)     1300 hours

C)     1500 hours

D)    1480 hours

13)  Consider a firm that uses bonuses to give its employees incentives to work hard. Suppose only 22% of employees get more than the average employee bonus. Which is a reasonable inference about the distribution of bonuses over employees based on this information?

A)   It is bimodal

B)   It is symmetric

C)   It is skewed to the left

D)   It is skewed to the right

14. 15. 16. 17. 18.

14) You constructed a 98% confidence interval for the population mean μ as [13, 17].  Consider the following two statements:

I.   Using the same data, the null hypothesis that μ = 16 would not be rejected by a 2-sided test at a significance level of 2%.

II.   Using the same data, the null hypothesis that μ = 19 would not be rejected by a 2-sided test at a significance level of 2%.

A)   Both statements are false.

B)   Both statements are true.

C)   Only statement I. is true.

D)   Only statement II. is true.

15) If Pr(A) is 0.4 and Pr(B) is 0.3, then Pr(A and B)

A)    equals 0.70

B)    cannot be determined with the information given

C)    equals 0.12

D)    equals 0.58

16) A large p-value for the null hypothesis H0 : μ  = μ0  implies

A)    that the observed sample average is consistent with the null hypothesis.

B)    a large observed sample average.

C)    the rejection of the null hypothesis.

D)    a large probability of type II error.

17) Which of the following is true about the Student-t distribution?

A)   As the sample size decreases, the Student-t distribution becomes more and more similar to the Standard Normal distribution

B) X-μ follows an exact Student-t distribution even if X is not normally distributed

C)   If the population standard deviation is known, we should find the critical values from the Student-t distribution table to compute confidence intervals for the population mean.

D)   If X is normally distributed, the sample size is small, and the population standard deviation is unknown, we should find the critical values from the Student-t distribution table to compute confidence intervals for the population mean.

18)  Suppose X is normally distributed with mean 10 and standard deviation 3. Then Y = 2X + 3 is

A)   normally distributed with mean 20 and standard deviation 6

B)   normally distributed with mean 20 and standard deviation 9

C)   normally distributed with mean 23 and standard deviation 6

D)   normally distributed with mean 23 and standard deviation 9

c.    (3 pts) If a larger sample had been used, would you expect the 99% confidence interval to be wider or narrower? Explain.

2)    (4 pts) A random sample of 142 college students was asked to indicate the number of hours per week that they surf the web for either personal information or material for a class assignment. The sample mean response was 6.06 hours and the sample standard deviation was 1.44 hours. Based on these results, a confidence interval extending from 5.86 to 6.16 was computed for

3)    (10 points) Granny Veggies is a producer of a wide variety of frozen vegetables. The company president has asked you to determine if the mean weekly sales of 16-oz packages of frozen broccoli has increased, after the implementation of an aggressive marketing campaign. In the past, the mean weekly sales per store of 16-oz packages of frozen broccoli is believed to have been 2,400 packages.

a.    (2 pts) State the null and the alternative hypotheses.

b.    (5 pts) You obtain a random sample of recent sales data from 138 stores for your study. The average weekly sales is 3,593 packages  and  the  sample  standard  deviation  is  4,919.  Compute  the  p-value  for  this  hypothesis  test  and  display  it graphically. What is your conclusion at the 1% significance level. Show all your work.

c.    (3  pts)  Suppose  that the alternative hypothesis had been two-sided, rather than one-sided.  State, without doing any calculations, whether the p-value of the test would be higher than, lower than, or the same as that found in part b. Sketch

4)    (11 points) During recent seasons, Major League Baseball (MLB) has been criticized for the length of the games. A report indicated that the average game lasted 195 minutes. The MLB has implemented changes that they believe have reduced the length of baseball games. You want to test this claim.

a.    (3 pts). State the null hypothesis and the alternative hypothesis of this test.

b.    (4 pts) You randomly select 86 games from the seasons after the changes were implemented and record their length. You decide that you will reject the null hypothesis if the average length of the games in your sample is less than 185 minutes. If that is your decision rule, what significance level do you have in mind? (Recall that the significance level is the probability of rejecting the null when the null is true). Assume that the population standard deviation is known and equal to 50 minutes. Show your work.

c.    (4 pts) If the average length of the games in your sample is 183 minutes, but you now want the significance level of your test to be 1%, what is your conclusion? Continue to assume that the population standard deviation is known and equal to

50 minutes. Show all your work.

5)    (4 pts) A statistics professor wants to compare today’s students with those 25 years ago. All his current students’ grades are stored on a computer so he can easily determine the population mean. However, the grades from 25 years ago reside in a cabinet, in moldy files.  He does not want to retrieve all the grades and will be satisfied with an estimate. He is planning on drawing a random sample of his former students’ grades to estimate the mean grade 25 years ago. How large a sample is needed to ensure that the probability that the sample mean is within 2 points of the true population mean is 95%? The professor assumes

6)    (8 points) Assume that the standard deviation of the time spent studying by students in the week before finals is 8 hours. A random sample of 100 students is taken to estimate the mean study time for the population of all students.

a.    (4 pts) What is the probability that that sample average is more than 1 hour below the population mean? Show and justify your work. In particular, justify the sampling distribution of the sample average you are using.

b.    (4 pts) Suppose that a second (independent) random sample of 150 students was taken. Without doing the calculations,

7)    (6 points) The restaurant in a large commercial building provides coffee for the building occupants. The restaurateur has determined over the years that the probability distribution of the number of cups of coffee consumed per day by the population in the building is:

The mean number of cups of coffee consumed in a day by the occupants in the building is 2. The variance is 0.8.

a.    (2 pts) We pick a person at random from the building. What is the probability that that person drinks more than 2 cups of coffee a day? Show your work.

b.    (4 pts) A new tenant of the building intends to have a total of 100 employees. Consider the 100 new employees as a random sample from a population with the same probability distribution as the one above. What is the probability that the total number

8)    (8 points)

a.    (4 pts) Let the random variables X1, X2, ⋯ , Xn   denote a random sample from a population with mean μ and variance σ 2 . Let

X- =  . Show that Var(X-)  . Show all the steps.

b.    (4 pts) Suppose the true population mean μ is 6. Consider the following four estimators of the population mean:

•    The estimator 1  that has a mean of 7

•    The estimator 2  that has a mean of 5

•     The estimator 3   = 0.5 1  + 0.5 2

•    The estimator 4   = 1  + 2 :

Which ones of these estimators are unbiased? For the ones that are biased, specify if the bias is positive or negative. Show your work.

9)    (12 points) An insurance company estimated that 30% of all automobile accidents were partly caused by weather conditions and that 50% of all automobile accidents involved bodily injury. Further, of those accidents that involved bodily injury, 20% were partly caused by weather conditions.

Define the following events: W = automobile accidents partly caused by weather conditions, and B = automobile accidents involving bodily injury.

a.    (3 pts) What is the probability that a randomly chosen accident both were partly caused by weather conditions and involved bodily injury? Show your work.

b.    (2 pts) Are the events W and B independent? Justify.

c.    (3 pts) If a randomly chosen accident was partially caused by weather conditions, what is the probability that it involved bodily injury? Show your work.

d.    (4 pts) What is the probability that a randomly chosen accident both was not partly caused by weather conditions and did not