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UNIVERSITY OF TORONTO Faculty of Arts and Science

APRIL 2007 EXAMINATIONS

ECO220Y1Y

PART 1 OF 2

Duration - 3 hours

Examination Aids: Calculator

This exam consists of two parts and a SCANTRON form. You may detach the formula sheets and statistical tables (Standard Normal distribution, Student t distribution, and F distribution) that are stapled to Part 2. Part 1: 4 written questions with point values next to each for a total of 45 points. If you run out of room you may continue your answers on pages 12 and 13, but clearly indicate you have done so (for example: “See page 12 for the rest of my answer…”) and clearly label your additional responses (for example: “Question (2) (b) continued:”). Part 2: 22 multiple choice questions worth 2.5 points each for a total of 55 points. Answers must be properly recorded on the SCANTRON form to earn any points. It is the student’s responsibility to hand in the completed SCANTRON form and all 13 pages of Part 1 of this exam. Any missing page will get zero marks. Last Name:

First Name:

Student ID #:

Check Your Section:

Mazaheri Murdock Yu L0101, L0201 L0301, L0401 L5101 M or W 11-1 T or R 11-1 T 6-9

Q1 Q2 Q3 Q4 Part 1 Part 2 Total

Point Value 12 12 12 9 45 55 100

Points Earned

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Problems: There are 4 problems with point values indicated next to each. Write your answers clearly, concisely, and completely on these examination papers. (1) [12 points] Consider the research question of whether woman exercise more regularly than men. A random sample of 200 women and 150 men yields these results:

Men Women

Exercise regularly 88 130

Do not exercise regularly 62 70

Total 150 200

(a) [3 points] Construct a 95% confidence interval estimate of the difference in the proportion of women and men who exercise regularly. Interpret the interval.

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(b) [5 points] Conduct a hypothesis test to determine if women exercise more regularly than men. For a 5% significance level, find the standardized rejection region AND the p-value. Make a conclusion for both the rejection region approach and p-value approach.

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(c) [4 points] If 65% of the women and 55% of the men exercise regularly, what is the power of your test in Part (b) for α = 0.05?

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(2) [12 points] A random sample of 62 seventh grade students is selected to study the relationship between Grade Point Average (GPA) and IQ (Intelligence Quotient). The following descriptive statistics are obtained:

GPA IQ Mean 7.5 105 Standard Deviation 2.1 16 The correlation coefficient between GPA and IQ is 0.64. (a) [4 points] Find the equation of a simple regression for predicting GPA from IQ. What is the interpretation of the intercept?

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(b) [5 points] Find the standard error of the slope estimate. Is the slope statistically significant at the 0.05 level?

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(c) [3 points] Suppose another variable measuring the number of courses a student is enrolled in is also included. The regression model is iiii XXY εβββ +++= 22110 , where Y is GPA, X1 is IQ, and X2 is the number of courses. Considering the information below, test the overall statistical significance of this model at the 0.05 significance level.

i YY ˆ− ( )2YY −

1 5.2 27.04

2 1.5 2.25

… … …

62 -3.1 9.61

Total 0 107.604

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X

Y

0

7

5

1

5

Group B

Group A (3) [12 points] Consider the graph. (a) [2 points] Write the equation for the line describing the relationship between Y and X for Group A. Write the equation for the line describing the relationship between Y and X for Group B. (b) [4 points] Suppose W is a variable that equals 1 for Group A and 0 for Group B. Write the equation for the estimated multiple regression model that is consistent with the graph. (Hint: The answer to this question is one equation, not two.)

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(c) [2 points] Write the theoretical model that corresponds to your estimated model in Part (b). Name the parameters β0, β1, … and use i as the observation index. (d) [4 points] How could you determine if there is any statistical difference between the line for Group A and Group B? Be specific and refer to your answers in Parts (b) and (c). Indicate what additional information you would need.

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(4) [9 points] David Naylor, the current president of U of T, has a mandate to “enhance the undergraduate experience.” Suppose that AAA Academic Consulting shows President Naylor an empirical study that it claims will “provide an obvious way to achieve the mandate.” AAA estimated the following regression.

STU_EXP = -0.280 + 2.341*GPA (0.092) (0.883) STU_EXP: Student’s experience on a scale from 0 to 10 GPA: Student’s grade point average on a 4 point scale (Standard errors are given in parentheses.)

The data are collected by randomly sampling 1,077 undergraduate students and asking: “Considering your experience as a student at U of T, how would you rate your experience on a scale from 0 to 10, where 0 is terrible and 10 is outstanding?” The GPA data are obtained directly from ROSI. (a) [5 points] Find the p-value for the test of a statistically positive slope. Explain the meaning of the p-value in one sentence. State the result of the statistical test.

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(b) [4 points] Does this analysis “provide an obvious way to achieve the mandate” as claimed by AAA Consulting? What can President Naylor conclude from this analysis?

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Extra Space: If you use this space, clearly indicate for which question(s).

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Extra Space: If you use this space, clearly indicate for which question(s).

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UNIVERSITY OF TORONTO Faculty of Arts and Science

APRIL 2007 EXAMINATIONS

ECO220Y1Y

PART 2 OF 2

Duration - 3 hours

Examination Aids: Calculator

This exam consists of two parts and a SCANTRON form. You may detach the formula sheets and statistical tables (Standard Normal distribution, Student t distribution, and F distribution) that are stapled to Part 2. Part 1: 4 written questions with point values next to each for a total of 45 points. Part 2: 22 multiple choice questions worth 2.5 points each for a total of 55 points. Answers must be properly recorded on the SCANTRON form to earn any points.

• Students that do not exactly follow the detailed SCANTRON INSTRUCTIONS on

the next page bear complete responsibility for any lost marks or a 0 mark.

Absolutely no remark requests related to the SCANTRON form will be

entertained for any student that does not completely follow the instructions.

• FILL IN ALL FIELDS ON THE TOP HALF OF THE PINK SCANTRON FORM

• Complete your SCANTRON form BEFORE the end of the exam is announced

o No extra time will be allowed It is the student’s responsibility to hand in the completed SCANTRON form.

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SCANTRON INSTRUCTIONS: ONLY those answers correctly marked on the SCANTRON form can earn marks

SCANTRON form must be completed BEFORE the end of the exam is announced

You MAY do scratch work on these pages

• Use only a pencil or blue or black ball point pen

• Pencil strongly recommended, it can be erased if a mistake is made

• Make dark solid marks that fill the bubble completely

• Erase completely any marks you want to change

o Crossing out a marked box is not acceptable and is scored as incorrect

• Select the one best alternative

• Correct answers are worth 2.5 points

• If you choose alternative (E) “Don’t know” you will be awarded partial credit of 0.65 points for being able to recognize that you don’t know the answer

• Incorrect answers are worth 0 points

1st : Print your LAST NAME and INITIALS in boxes provided Use exact name you are officially registered under Darken each letter in the corresponding bracket below each box

2nd : Print your 9 digit STUDENT NUMBER in the boxes provided Fill in zeros in front of the number if less than 9 digits Darken each number in the corresponding bracket below each box

3rd : Print 2 digit FORM number in the boxes provided Your FORM number is 01 Darken each number in the corresponding bracket below each box

4th : Sign your name in the SIGNATURE box

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Multiple Choice Questions: There are 22 questions. For each, choose the best answer and mark it on the SCANTRON form. Correct answers worth 2.5 points each. “Don’t know,” choice (E), worth 0.65 points each. Incorrect answers worth 0 points each. (1) An elevator in a hotel can safely carry a total weight of up to 5400 pounds. The average weight of guests is 130 pounds with a standard deviation of 60 pounds. Suppose 36 guests get into the elevator. Assuming the weights of these guests are independent, what is the chance of an unsafe weight in the elevator?

(A) 0.0013 (B) 0.0175 (C) 0.0228 (D) 0.0312 (E) Don’t know

(2) The distribution of X is bell shaped. If 95% of the observations are between 68 and 132, what is the standard deviation of the distribution?

(A) 8 (B) 11 (C) 16 (D) 32 (E) Don’t know

► For Question (3), consider this tabulation of a random sample of the variable Z.

Z | Freq. Percent Cum. ------------+----------------------------------- 9 | 12 2.15 2.15 10 | 41 7.35 9.50 11 | 201 36.02 45.52 12 | 304 54.48 100.00 ------------+----------------------------------- Total | 558 100.00

(3) What is the sample median?

(A) 10.5 (B) 11.0 (C) 11.5 (D) 12.0 (E) Don’t know

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► For Question (4), consider this histogram of a random sample of the variable X.

0

.01

.02

.03D

ensi

ty

0 10 20 30 40x

(4) About what percent of observations are between 5 and 10?

(A) 2.0% (B) 10.0% (C) 12.5% (D) 15.0% (E) Don’t know

► For Questions (5) – (6), consider a Binomially distributed random variable X where the probability of success is 0.15. (5) If the number of trials is 10, what is the probability that the number of successes is within one standard deviation of the mean?

(A) 62% (B) 64% (C) 66% (D) 68% (E) Don’t know

(6) If the number of trials is 1000, what is the probability that the number of successes is within one standard deviation of the mean?

(A) 62% (B) 64% (C) 66% (D) 68% (E) Don’t know

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(7) A corporation produces packages of paper clips. The number of clips per package is random with a mean of 80 and a standard deviation of 2.4. The cost of producing a package of clips is (0.10 + 0.015*X) dollars, where X is the number of clips in the package. The revenue from selling the package, however many clips are in it, is $1.50. If profit is defined as the difference between revenue and cost, what is the mean and standard deviation of profit per package in dollars?

(A) 0.20 and 0.018 (B) 0.20 and 0.036 (C) 0.25 and 0.018 (D) 0.25 and 0.036 (E) Don’t know

(8) Given a random sample where the Central Limit Theorem holds, which statements about X and X are true?

I. XX =

II. ][][ XEXE =

III. ][][ XVXV =

(A) I. (B) II. (C) II. and III. (D) I., II. and III. (E) Don’t know

(9) Which of the following would increase the power of a hypothesis test for the population proportion?

(A) Reduce the sample size (B) Reduce the significance level from 0.05 to 0.01 (C) Use economic intuition to make a two-tailed test into a one-tailed test (D) Change the null hypothesis so that it is closer to the true population proportion (E) Don’t know

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(10) For which of the following hypotheses tests would the p-value be the same whether the sample mean is 44 or 46?

I. H0: μ = 45 H1: μ ≠ 45

II. H0: μ = 45 H1: μ > 45

III. H0: μ = 45 H1: μ < 45

IV. H0: μ = 46 H1: μ ≠ 46

(A) I. (B) I. and IV. (C) II. and III. (D) IV. (E) Don’t know

► For Questions (11) – (12), suppose a company wishes to estimate the proportion of defective items in a production line. (11) To estimate the proportion within 0.02 and with 98% confidence, what is the minimum required sample size if the company currently has no idea about the proportion defective? (A) 2401

(B) 3394 (C) 4129 (D) 4302 (E) Don’t know

(12) Suppose a random sample of 400 items contains 250 defective items. What is the 98% confidence interval estimate of the proportion defective? (A) (0.5686, 0.6814) (B) (0.5711, 0.6789) (C) (0.6050, 0.6450) (D) (0.6088, 0.6412)

(E) Don’t know

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► For Questions (13) – (16): A firm decides to launch a new product only if it can infer that the mean number of units purchased per customer in the first solicitation is more than 2.0. The null and alternative (research) hypotheses are H0: μ = 2 and H1: μ > 2. For the number of units purchased per customer in the first solicitation, μ is the mean and 1.6 is the population standard deviation. A random sample of 100 customers is selected. (13) If the sample mean is over 2.2496, the firm will launch the product. Otherwise they will not launch the product. What is the probability of making a Type I error?

(A) 0.0015 (B) 0.0122 (C) 0.0244 (D) 0.0594 (E) Don’t know

(14) If the firm wishes to set the probability of a Type I error at 0.01, what values of the sample average would lead it to launch the product?

(A) X > 2.0000 (B) X > 2.2632 (C) X > 2.3136 (D) X > 2.3728 (E) Don’t know

(15) If X is 2.4 units, what is the p-value?

(A) 0.0003 (B) 0.0027 (C) 0.0062 (D) 0.0100 (E) Don’t know

(16) If μ is 2.15 and α is 0.01, what is the probability of a Type II error?

(A) 0.0823 (B) 0.5823 (C) 0.9177 (D) 0.9900 (E) Don’t know

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► For Question (17): Consider the following hypothesis test: H0: μ = 20 H1: μ > 20 Suppose that μ is 21, n is 100, and the significance level is 0.05. (17) Which of the following would decrease the chance of making a Type II error?

(A) The hypotheses specify μ = 19 and μ > 19 (instead of μ = 20 and μ > 20) (B) The true population mean is 20.5 (instead of 21) (C) The significance level is 0.01 (instead of 0.05) (D) n is 50 (instead of 100) (E) Don’t know

► For Questions (18) – (19), consider the research question of whether female drivers drive faster than male drivers. 63 female drivers and 47 male drivers are randomly selected. For each the highway driving speed (mph) is recorded. For female drivers,

671 =X and 7.41 =s . For male drivers 652 =X and 8.42 =s . Assume 22

21 σσ = and

that speeds are normally distributed. (18) What is the p-value for the implied hypothesis test?

(A) p-value < 0.010 (B) 0.010 ≤ p-value < 0.025 (C) 0.025 ≤ p-value < 0.050 (D) p-value ≥ 0.050 (E) Don’t know

(19) Suppose you use a linear regression to answer the question. The sample size is 110. If MPH records speed, FEM = 1 if female driver, 0 otherwise, MAL = 1 if male driver, 0 otherwise, which of the following would be the estimated regression?

(A) FEMHPM *265ˆ += (B) FEMHPM *269ˆ −= (C) MALHPM *263ˆ += (D) MALHPM *269ˆ −= (E) Don’t know

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► For Questions (20) – (22), consider a regression model to explain the number of students enrolled in colleges. Explanatory variables include the percent female enrollment (PercFem), the percent of students from within province (PercProv), the rank of the college (Tier_Two = 1 if in second tier, 0 otherwise; Tier_Three = 1 if in third tier, 0 otherwise), whether the college offers a business degree (Business = 1 if business degree is offered, 0 otherwise), and whether the college is located in the East (East = 1 if in the East, 0 otherwise). The sample size is 93.

Coefficient SEIntercept 2013.0 227.5PercFem 6.13 2.60PercProv -3.82 1.12Tier_Two -156.5 63.5Tier_Three -307.2 105.7Business 163.8 68.0East 350.9 445.9

(20) The coefficient on which variable is NOT statistically significant?

(A) PercFem (B) PercProv (C) Tier_Two (D) East (E) Don’t know

(21) What is the predicted enrollment for a first tier college in the East with 60 percent females and 40 percent from within province that does not offer a business degree?

(A) 566 (B) 943 (C) 1,936 (D) 2,579 (E) Don’t know

(22) What is the predicted difference in enrollment between a comparable second tier and third tier college?

(A) The second tier college is predicted to enroll 151 more students (B) The second tier college is predicted to enroll 157 less students (C) The second tier college is predicted to enroll 307 more students (D) The second tier college is predicted to enroll between 157 and 307 more students (E) Don’t know