If a mound-shaped distribution is symmetric ... - Illinois



Midterm I—Form A Economics 203 Name_____________

Fall 2004 Instructor: Petry SSN______________

Before beginning the exam, please verify that you have 13 pages with 40 questions in your exam booklet. You should also have a decision-tree and formula sheet provided by your TA. Please include your full name and Net-ID on your bubble sheets. Good luck!

1. Which of the following statements is NOT true concerning confidence intervals?

a. there are always two tails outside the confidence interval

b. they always contain the relevant population parameter

c. they always contain the relevant point estimate

d. they can sometimes be used for hypothesis testing

e. all of the above statements are true of confidence intervals

2. Which of the following is always true?

a. the covariance of a variable with itself is always equal to one

b. the coefficient of correlation of a variable with itself is always equal to one

c. the covariance between two variables is always positive

d. the correlation coefficient between two variables is always positive

e. the covariance between two variables can never be zero

3. The math placement exam scores for freshmen at the University of Illinois are normally distributed with mean 75 and standard deviation 10. What is the probability of seeing a score greater than 90 or a score less than 60? (Note: you should rely on the probabilities given below to answer this question.)

| (0 < z < 1.5) =0 .433 |P (-2.5< z < 0) =0 .494 |

a. 0.067

b. 0.062

c. 0.134

d. 0.124

e. 0.866

4. Consider a positively skewed mound-shaped distribution. Which of the following is true?

a. mean < median < mode

b. median < mode < mean

c. mode < mean < median

d. mode < median < mean

e. median < mean < mode

Use the following information to answer the next two questions (#5-6)

A test preparation service claims that at least 10% of the students who take their course get a perfect score on the math section of the GRE. To investigate this claim, you sampled 200 people who used this service and found that 28 got a perfect score on the math section of the GRE.

5. What is the appropriate test statistic for testing the company’s claim?

z0.1=1.28 z0.05=1.645 z0.025=1.96

a. 1.417

b. 1.558

c. 1.630

d. 1.886

e. 2.074

6. Ignoring the result of the previous question, assume that the correct test statistic was 1.659. Suppose that you want the probability of Type I error to be no more than .05. What should you conclude in this case?

a. fail to reject the null hypothesis and conclude that the test preparation company can claim that at least 10% of their students get a perfect score on the math section

b. fail to reject the null hypothesis and conclude that the test preparation company cannot claim that at least 10% of their students get a perfect score on the math section

c. reject the null hypothesis and conclude that the test preparation company can claim that at least 10% of their students get a perfect score on the math section

d. reject the null hypothesis and conclude that the test preparation company cannot claim that at least 10% of their students get a perfect score on the math section

e. there is not enough information to say anything informative about the claim

7. Suppose you’re considering whether to pursue a PhD in economics at the University of Illinois, and you want to know whether being a graduate student in economics at Illinois is really as much fun as you’ve heard it is. You interview 120 people who were graduate students in economics at Illinois, and find that 80 of them thought it was really fun. A 95% confidence interval for the share of graduate students who thought it was really fun would be:

z0.1=1.28 z0.05=1.645 z0.025=1.96

a. [0.524, 0.809]

b. [0.582, 0.751]

c. [0.507, 0.826]

d. [0.466, 0.867]

e. [0.422, 0.911]

8. Chicago Cubs, a baseball team, is considering trading Sammy Sosa because he has not been hitting as many home runs as the management would like. The club will buy a new player only if they find one that can hit at least 10 more home runs than Sammy Sosa hits in a season. They have their eyes on Gary Sheffield of the New York Yankees. The Cubs management gathers data on 18 seasons that Sammy Sosa has played, and during that period he hit an average of 17 home runs per season with the standard deviation of 12, while is his 19 years of play, Gary Sheffield hit an average of 27 home runs per season with the standard deviation of 12. What decision should they make?

a. reject the null hypothesis and trade Sammy Sosa for Gary Sheffield

b. reject the null hypothesis and do not trade Sammy Sosa for Gary Sheffield

c. fail to reject the null hypothesis and trade Sammy Sosa for Gary Sheffield

d. fail to reject the null hypothesis and do not trade Sammy Sosa for Gary Sheffield

e. there is not enough information to say anything informative about the claim

Use the following information to answer the next two questions (#9-10)

One of the teaching assistants claims that the grades on the first exam last semester were closer together than they had been in the past. To test if this claim is true, he looks at a sample of 12 exams from last semester:

44, 45, 44, 46, 47, 46, 47, 49, 44, 42, 48, 43

Historically, the variance of exam scores was 4.232.

|(20.975,11 = 3.82 |(20.025,11 = 21.92 |(20.975,12 = 4.40 |(20.025,12 = 23.34 |

9. What is the value of the appropriate test statistic for examining the TA’s claim?

a. 9.415

b. 9.910

c. 10.432

d. 10.981

e. 11.559

10. What is the lower level of a 95% confidence interval for the variation in grades last semester?

a. 2.096

b. 2.232

c. 4.076

d. 12.805

e. 11.117

Use the following information to answer the next two questions (#11-12)

You are an analyst for a shipping company and you are asked to determine whether workers who have been on the job more than 20 years have different productivity than workers in general. Data is collected on how many packages a day are shipped from a random sample of 50 workers who have been on the job more than 20 years. The mean and standard deviation calculated from this sample are 267 and 45, respectively. From other research, you know that the average for workers in general is 280.

11. What can you say about the test statistic?

a. it equals 2.043 and follows a t distribution with 49 degrees of freedom

b. it equals 2.043 and follows a t distribution with 50 degrees of freedom

c. it equals -2.043 and follows a z distribution with 49 degrees of freedom

d. it equals -2.043 and follows a t distribution with 49 degrees of freedom

e. it equals -2.043 and follows a t distribution with 50 degrees of freedom

12. If you were testing this hypothesis at the 1% significance level, what would you conclude?

z0.05=1.645 t0.05,49= 1.68 t0.01,49 = 2.40

z0.025= 1.96 t0.025,49 =2.01 t0.005,49=2.68

a. fail to reject the null hypothesis and conclude there is not enough evidence to claim that there is a difference in the productivity of older workers

b. reject the null hypothesis and conclude there is not enough evidence to claim that there is a difference in the productivity of older workers

c. fail to reject the null hypothesis and conclude that older workers are less productive

d. reject the null hypothesis and conclude that older workers are less productive

e. reject the null hypothesis and conclude that productivity is different for older workers

13. One of your fellow students suspects that Canadians spend more time reading newspapers than Americans. To test her claim, she gathers a sample of the average number of hours per week spent reading newspapers for 100 students at the University of Saskatoon and does the same for 100 students at the University of Illinois. Assuming that her choice of tests is limited to the following, which test should she use to examine her claim?

a. Z-test for difference in proportions

b. Z-test for difference in means

c. paired sample t-test for difference in means

d. t-test for difference in means assuming unequal variances

e. F-test for difference in variances

14. Which of the following statements is true?

a. in a test on the difference of means between two populations, which population we choose as population one will not affect the relevant point estimate

b. in a test on the difference of means between two populations, which population we choose as population one will not affect the test statistic

c. in a test on the difference of means between two populations, which population we choose as population one will not affect the critical value

d. in a test on the difference of means between two populations, which population we choose as population one will not affect the conclusion of the test

e. all of the forgoing statements are false

15. Which of the following is implied by the Central Limit Theorem?

a. if a random sample is drawn from any population then the sampling distribution of the sample mean is approximately normal for any sample size

b. if a random sample is drawn from any population then the sampling distribution of the sample mean is approximately normal for a sufficiently large sample size

c. if a random sample is drawn from any population then the sampling distribution of the sample mean is approximately standard normal for any sample size

d. if a random sample is drawn from any population then the sampling distribution of the sample mean is approximately standard normal for a sufficiently large sample size

e. if a random sample is drawn from a normal population then the sampling distribution of the sample mean is approximately normal for a sufficiently large sample size

16. Suppose you are a Congressional candidate trying to allocate resources between advertising in the Daily Illini or in the student newspaper at Illinois State. You think that the Daily Illini reaches fewer students. To test this claim, which could be a proper null and alternative hypotheses (data is the form of the average number of students who read the Daily Illini followed by number of students who read the ISU paper on 25 randomly selected days during the academic year)?

a. H0: μ1 ≥ μ2; H1: μ1 ≠ μ2

b. H0: μ1 = μ2; H1: μ1 ≤ μ2

c. H0: μ1 = μ2; H1: μ1 < μ2

d. both a and c

e. a, b, and c

Use the following information to answer the next four questions (#17-20)

You are enrolled in a gender studies class at the UIUC and you professor has given you an assignment to test a claim by a renowned researcher who claims that female students spend more money on textbooks than male students. You do a survey on campus and gather data on 10 female (sample 1) and 12 male (sample 2) students. Female students spend, on average, $450 per semester on books, with a standard deviation of $75. Male students, on the other hand, spend an average of $405 on books per semester, with a standard deviation of $45. Moreover, from previous studies you know that variances of the two populations are the same.

17. To test the researcher’s claim, you would perform a certain hypothesis test. What would your null and alternative hypotheses look like?

a. H0: μ1- μ2 ≥ 0; H1: μ1- μ2 < 0

b. H0: μ1- μ2 ≤ 0; H1: μ1- μ2 > 0

c. H0: [pic] ≤ 1; H1: [pic]> 1

d. H0: [pic] ≥ 1; H1: [pic]< 1

e. H0: [pic] ≤ 0; H1: [pic]> 0

18. What would be the value of the point estimate?

a. 1.667

b. 1.741

c. 13.416

d. 30

e. 45

19. Ignoring the answer that you got in the previous question, suppose that the correct test statistic was 2.101. What is your conclusion about the researcher’s claim at the 5% level of significance?

t0.05,14=1.761 t0.025,14=2.148 t0.01,14=2.624

t0.05,20=1.725 t0.025,20=2.086 t0.01,20=2.528

a. reject the null hypothesis and conclude that female students spend less on textbooks than male students

b. fail to reject the null hypothesis and conclude that female students spend more on textbooks than male students

c. reject the null hypothesis and conclude that male students spend more on textbooks than female students

d. reject the null hypothesis and conclude that female students spend more on textbooks than male students

e. fail to reject the null hypothesis and conclude that there is insufficient evidence to conclude that female students spend more on textbooks than male students

20. Now suppose that you are skeptical about the equal population variances assumption used throughout this set of questions. So, you perform an appropriate test. What is your conclusion at the 10% level of significance?

F0.025, 9, 11=3.588 F0.025, 10, 12=3.374 F0.05, 9, 11=2.896

F0.05,10, 12=2.754 F0.1, 9, 11=2.274 F0.1, 10, 12=2.188

a. there is insufficient evidence to conclude that the population variances are equal, so the above assumption of equal variances is correct

b. there is insufficient evidence to conclude that the population variances are equal, so I would have to re-evaluate the above answers

c. there is enough evidence to conclude that the population variances are unequal, so the above assumption of equal variances is correct

d. there is enough evidence to conclude that the population variances are unequal, so I would have to re-evaluate the above answers

e. there is insufficient evidence to conclude that the population variances are unequal, so the above assumption of equal variances is correct

21. The weights of the Illini football team were collected and it was found that the mean weight was 230 pounds with a standard deviation of 25 pounds. Approximately, what fraction of the team weighs more than 305 pounds?

a. 99%

b. 5%

c. 0.005%

d. 0.5%

e. -98.5%

22. An education researcher is interested in the distribution of tuition bills for private schools in Chicago. She collects the following data on private schools in Chicago and reported the following values in thousands of dollars.

|School |

| | | |

|  |Widgets |Snidgets |

|Mean |96.66 |101.39 |

|Variance |314.512 |354.75 |

|Observations |10 |10 |

|Hypothesized Mean Difference |0 | |

|df |18 | |

|t Stat |-0.578 | |

|P(T 0.1 than if you use H1: σ2 ≠ 0.1

b. it will require more extreme values of the test statistic to reject the null hypothesis if you use H1: σ2 ≠ 0.1 than if you use H1: σ2 > 0.1

c. the choice of H1: σ2 > 0.1 versus H1: σ2 ≠ 0.1 will not affect how extreme a test statistic you will need to reject the null hypothesis

d. you do not have enough information to make any statement about the effect of choosing H1: σ2 > 0.1 versus H1: σ2 ≠ 0.1 on how extreme a test statistic you will need to reject the null hypothesis

Use the following information to answer the next two questions (#27-28)

Your high-school friend claims that the Northwestern football team has been fairly consistent in the last 20 years in terms of the numbers of wins each season. Illinois, he adds, has been more unpredictable. There are years when Illinois would win almost all the games, but there were seasons where it would win only 2 games per season. So, you decide to gather data and test your friends claim that the Northwestern football team (sample 1) has been more consistent than the Illinois football team (sample 2) in the past 20 years. The table below contains the summary statistics of your data.

| |Northwestern |Illinois |

|Mean |5 |7 |

|Standard deviation |14 |20 |

|Number of observation |20 |20 |

24. Given that the relevant critical value is 0.421, what would be your test statistic and the conclusion for this test?

a. -2.317 and you would reject the null hypothesis

b. -0.366 and you would fail to reject the null hypothesis

c. 0.366 and you would reject the null hypothesis

d. 0.490 and you would fail to reject the null hypothesis

e. 0.490 and you would reject the null hypothesis

25. If, on the other hand, you wanted to test whether Illinois has been a better team (more wins) than Northwestern in each of the past 20 years, which test would you perform?

a. t-test for difference in means assuming equal variances

b. Z-test for difference in means

c. Z-test for difference in proportions

d. F-test for difference in variances

e. paired sample t-test for difference in means

Use the following information to answer the next two questions (#29-30)

You just moved to London and since you are a big soccer fan, you would like to choose a local team that you will root for. You live close to the Chelsea FC stadium, but the problem is that you would like to be a fan of a team that wins most of the time. So, although Chelsea’s stadium is closer than Arsenal’s, you are willing to sacrifice and become an Arsenal fan if Arsenal wins more than 10% more than Chelsea. Last year, Arsenal (sample 1) won 42 out of 56 games, while Chelsea (sample 2) won 30 out of 50 games.

26. What is the relevant point estimate?

a. 0.05

b. 0.15

c. 0.554

d. 0.679

e. 1.662

27. Ignoring the results of the previous question, suppose that the test statistic is -1.735. What is the p-value for the above test?

z0.1=1.28 z0.05=1.645 z0.025=1.96

a. less than 0.025

b. less than 0.05 but more than 0.025

c. less than 0.5 but more than 0.05

d. less than 0.95 but more than 0.5

e. less than 0.975 but more than 0.95

28. You want to find out whether the average income in Champaign is different than the average income in Urbana. You gather data and the average income in Champaign (sample 1) is $36,900 with standard deviation $14,200, while the average salary in Urbana (sample 2) is $39,100 with standard deviation $16,800. You calculate the 95% confidence interval for the true difference in population means and it turns out to be [-5543.453, 1143.451]. How do you proceed?

a. no data is provided on the numbers of observations for each sample, so I cannot proceed

b. I cannot determine whether I should perform an equal or unequal variances t-test for difference in means since I cannot perform a F-test for the ratio of variances

c. I would reject the null hypothesis and conclude that Champaign and Urbana have the same average income

d. I would fail to reject the null hypothesis and conclude that there is insufficient evidence to conclude that average incomes are different in Champaign and Urbana

e. I would reject the null hypothesis and conclude that the average income in Urbana is higher than in Champaign

Use the following information to answer the next two questions (#32-33)

Suppose that you want to investigate a relationship between the years of schooling and salary. You gather data on 6 people and the results are presented in the following table:

|Years |12 |14 |12 |16 |18 |20 |

|Salary($1000) |16 |17.5 |22 |35 |62 |42 |

29. Calculate the sample covariance and correlation coefficient between years of schooling and salary.

a. cov(x,y) = 39.1, r = .810

b. cov(x,y) = .810 , r = 46.933

c. cov(x,y) = 46.9 , r = .810

d. cov(x,y) = 46.9 , r = .889

e. none of the above

30. Using the answers from the previous question, which of the following conclusions can you make about the relationship between years of schooling and salary?

a. the covariance value tells us that there is a strong positive linear relationship between years of schooling and salary

b. the correlation coefficient tells us that the slope of the line that defines the relationship between the two variables is positive

c. the correlation coefficient is the value of the slope of the line that defines the relationship between the schooling and salary

d. both a and b

e. all of the above

31. In testing the difference between two population means assuming unequal variances, your p-value was larger than .05, so you did not reject the hypothesis that the difference in means was zero. You report that you are 95% confident that the means are equal. Was this correct?

a. yes, this it what it means to be 95% confident

b. no, you do not have the basis to make this claim

c. it would be correct, if you knew that the population variances were equal.

d. it would be correct, if the population variances were known

e. regardless of what you know about the population variances, you do not have enough information to make any statement about the relationship between the population means

32. Prof. Petry claims that people who go to the review session before the exam do better on the exam than people who don’t go to the review session. To test his claim, Marcus collected data on 75 students who went to the review session (sample 1) and 58 students who did not go to the review session (sample 2). The mean score of the students who went to the review session was 80 with the standard deviation of 12, while the mean score of the students who did not go to the review session was 84 with the standard deviation of 12. From the tests listed below, which would be the most appropriate for Marcus to perform?

a. t-test for difference in means assuming equal variances

b. Z-test for difference in means

c. Z-test for difference in proportions

d. (2-test for difference in variances

e. paired sample t-test for difference in means

33. All of the following statements are false, except

a. the variance of the sampling distribution of the sample mean is always equal to the variance of the population

b. the sample standard deviation is the square root of the sum of squared deviations from the population mean divided by the sample size minus one

c. the range is a measure of central tendency

d. the median is the value that falls in the middle of a set of ordered measurements

e. both b and d

34. Suppose that you want to estimate what share of the students got more than 70 points on their Econ 203 midterm exam. Moreover, you want your estimate to be within (0.1 of its true population value with 90% confidence. The smallest sample size required to accomplish this is:

z.1 = 1.28 z.05 =1.645 z.025 = 1.96

a. 40

b. 41

c. 59

d. 67

e. 68

35. Females on average spend more on clothing apparel than males. You want to find out whether more females spend at least 1/3 of their income on clothing and apparel than males do. Which of the following tests can you use?

a. pooled-variance t-test for difference in means

b. paired sample t-test for the mean difference

c. Z-test for difference in proportions

d. F-test for difference in variances

e. F-test for difference in means

36. Given the following p-values and their associated Z-scores, what is the last entry in the table?

|p-value |Z-statistic |

|0.023 |2 |

|0.050 |1.645 |

|0.025 |1.96 |

|0.159 |1 |

|0.284 |0.57 |

|0.097 | |

a. 0.3

b. 0.8

c. 1.3

d. 1.8

e. 2.3

37. To test whether more than 50% of the UIUC students spend 40% or more of their free time watching TV, your null hypothesis would be:

a. H0: p ≤ 0.40

b. H0: p > 50

c. H0: p = 0.40

d. H0: p ≤ 0.50

e. H0: p ≥ 0.50

ANSWER KEY

|1. B | |

|2. B | |

|3. C | |

|4. D | |

|5. D | |

|6. C | |

|7. B | |

|8. D | |

|9. E | |

|10. B | |

|11. D | |

|12. A | |

|13. D | |

|14. D | |

|15. B | |

|16. C | |

|17. B | |

|18. E | |

|19. D | |

|20. E | |

|21. C or D | |

|22. E | |

|23. D | |

|24. C | |

|25. D | |

|26. B | |

|27. D | |

|28. A or E | |

|29. B | |

|30. E | |

|31. D | |

|32. C | |

|33. B | |

|34. B | |

|35. A | |

|36. D | |

|37. E | |

|38. C | |

|39. C | |

|40. D | |

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