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ECON 2142 Graded Homework 2 Answer the questions in the spaces provided. If you run out of room for an answer, continue on a new sheet of paper stapled to the end of this packet. Name: 1. (6 points) A trucking company has found that its trucks average 0.2 breakdowns during round trips from New York to Los Angeles. (a) (2 points) What is the probability that a single truck will make the complete trip without experiencing a breakdown? (b) (2 points) If 3 trucks are assigned to a NY/LA round trip, what is the probability that at least 2 of them will make the complete trip without experiencing a breakdown? (c) (2 points) In general, how are the binomial and Poisson distributions related? ECON 2142 Graded Homework 2 2. (5 points) The campaign manager for a political candidate claims that 55% of registered voters favor the candidate over her strongest opponent. Assume that this claim is true and that we’re interested in computing the probability that in a simple random sample of 300 voters, at least 60% would favor the candidate over her strongest opponent. (a) (1 point) What are the values of n and ⇡ to be used in the sampling distribution of the proportion? A. B. C. D. ⇡ ⇡ ⇡ ⇡ = 0.40 = 0.45 = 0.55 = 0.60 and and and and n = 300 n = 300 n = 300 n = 300 (b) (1 point) What is the standard error of the sampling distribution of the proportion? Recall that you need to use ⇡ if it’s available in the calculation of p . A. B. C. D. p p p p ⇡ 0.03 ⇡ 0.04 ⇡ 0.49 ⇡ 0.50 (c) (1 point) What is the probability that at least 60% would favor the candidate over her strongest opponent? A. B. C. D. P (p P (p P (p P (p 0.60) ⇡ 0.0062 0.60) ⇡ 0.0475 0.60) ⇡ 0.1056 0.60) ⇡ 0.4594 (d) (1 point) Suppose that only 49% of the registered voters in the sample favor the candidate over her strongest opponent. If the campaign manager’s claim of 55% is correct, what is the probability that the sample proportion would be no more than 0.49 for the same sample size of 300? A. B. C. D. P (p  0.49) ⇡ 0.0001 P (p  0.49) ⇡ 0.0030 P (p  0.49) ⇡ 0.0228 P (p  0.49) ⇡ 0.0668 (e) (1 point) Based on your answer to part (d), speculate on whether the campaign manager’s claim might be mistaken. A. B. C. D. The The The The campaign manager is most likely correct in his claim. campaign manager is most likely mistaken in his claim. sample data is most likely correct. sample date is most likely mistaken. Page 2 ECON 2142 Graded Homework 2 3. (6 points) Historically, Shop-Mart has gotten an average of 2000 hours of use from its G&E fluorescent lightbulbs. Because its fixtures are attached to the ceiling, the bulbs are rather cumbersome to replace, and Shop-Mart is looking into the possibility of switching to Phipps bulbs, which cost the same. A sample of Phipps bulbs lasted an average of 2080 hours, and the p-value in a right-tail test (H0 : µ  2000 versus H1 : µ > 2000) is 0.012. Shop-Mart has shared these results with both G&E and Phipps. (a) (1.5 points) Give Shop-Mart a recommendation. In doing so, use “0.012” in a sentence that would be understood by a Shop-Mart executive who has had no statistical training but is comfortable with probabilities as used by weather forecasters. (b) (1.5 points) In interpreting these results, what level of significance might G&E like to use in reaching a conclusion? In other words, for what level of ↵ would a G&E executive like to hold the test to reach his desired outcome in the hypothesis test? (c) (1.5 points) In interpreting these results, what level of significance might Phipps like to use in reaching a conclusion? In other words, for what level of ↵ would a Phipps executive like to hold the test to reach his desired outcome in the hypothesis test? (d) (1.5 points) If the test had been two-tail (i.e., H0 : µ = 2000 versus H1 : µ 6= 2000) instead of right-tail, would the p-value still be 0.012? If not, what would the p-value be? Explain. Page 3 ECON 2142 Graded Homework 2 4. (5 points) Suspecting that television repair shops tend to charge women more than they do men, Emily disconnected the speaker wire on her portable television and took it to a sample of 12 shops. She was given repair estimates that averaged $85, with a standard deviation of $28. Her friend John, taking the same set to another sample of 9 shops, was provided with an average estimate of $65, with a standard deviation of $21. Assuming normal populations with equal standard deviations, use the 0.05 level in evaluating Emily’s suspicion. The following multiple choice questions are geared towards evaluating this hypothesis test. (a) (1 point) What is the two-sample hypothesis test we must use for this question? A. B. C. D. z-test for Independent Samples Unequal variances t-test Pooled-variances t-test Comparing Proportions from Independent Samples (b) (1 point) Suppose µ1 denotes the population mean for women and µ2 denotes the population mean for men. What are the null and alternative hypotheses? A. B. C. D. H0 H0 H0 H0 : µ1 = µ2 versus H1 : µ1 µ2 versus H1 : µ1  µ2 versus H1 : µ1 6= µ2 versus H1 : µ1 : µ1 : µ1 : µ1 6= µ2 < µ2 > µ2 = µ2 (c) (1 point) What is the value of s2p ? A. B. C. D. 10.69 25.29 114.33 639.58 (d) (1 point) If your answer to part (a) is a t test, what are tcalculated and tcritical , respectively? Alternatively, if your answer to part (a) is a z test, what are zcalculated and zcritical , respectively? A. B. C. D. 1.729; 1.793; 2.093; 1.793; 1.793 1.729 1.793 2.093 (e) (1 point) What is the conclusion at the 0.05 significance level? What can we say about the p value for this test? A. B. C. D. Reject H0 ; the p Reject H0 ; the p Fail to reject H0 ; Fail to reject H0 ; value value the p the p is > 0.05 is < 0.05 value is > 0.05 value is < 0.05 Page 4 ECON 2142 Graded Homework 2 5. (5 points) As a follow-up to the previous question, using the 0.05 level of significance in comparing the sample standard deviations, were we justified in assuming that the population standard deviations were equal? Would your conclusion change if the standard deviation of the estimates received by Emily had been $35? (Hint: you need to run two hypothesis tests of whether the population standard deviations are equal using the original value of $28 and the new value of $35) Page 5 ECON 2142 Graded Homework 2 6. (5 points) A pharmaceutical manufacturer has come up with a new drug intended to provide greater headache relief than the old formula. Of 250 patients treated with the previous medication, 130 reported “fast relief from headache pain.” Of 200 individuals treated with the new formula, 128 said they got “fast relief.” At the 0.05 level, can we conclude that the new formula is better than the old? Using the appropriate statistical table, what is the approximate p-value for this test? The following multiple choice questions are geared towards evaluating this hypothesis test. (a) (1 point) What is the two-sample hypothesis test we must use for this question? A. B. C. D. z-test for Independent Samples Unequal variances t-test Pooled-variances t-test Comparing Proportions from Independent Samples (b) (1 point) Suppose ⇡1 denotes the proportion of relieved individuals using the old medication and ⇡2 denotes the proportion of relieved individuals using the new medication. What are H0 and H1 for this hypothesis test? A. B. C. D. H0 H0 H0 H0 : ⇡1 ⇡2 versus : ⇡1 > ⇡2 versus : ⇡1 = ⇡2 versus : ⇡1 < ⇡2 versus H1 H1 H1 H1 : ⇡1 < ⇡2 : ⇡1  ⇡2 : ⇡1 6= ⇡2 : ⇡1 ⇡2 (c) (1 point) What is the value of p̄? A. B. C. D. p̄ = 0.57 p̄ = 1.16 p̄ = 225 p̄ = 450 (d) (1 point) What is the relevant test-statistic, i.e. zcalculated or tcalculated ? A. B. C. D. -2.56 -1.65 0.05 0.57 (e) (1 point) What is the conclusion for this hypothesis test? A. B. C. D. Fail to reject H0 and conclude that the old formula is better than the new formula Reject H0 and conclude that the old formula is better than the new formula Reject H0 and conclude that the new formula is better than the old formula Fail to reject H0 and conclude that the new formula is better than the old formula Page 6 ECON 2142 Graded Homework 2 7. (6 points) The idling speed of 14 gasoline-powered generators is measured with and without an oil additive that is designed to lower friction. With the additive installed, the mean change in speed was 123 revolutions per minute (rpm), with a standard deviation of 3.5 rpm. At the 0.05 level of significance, is the additive e↵ective in increasing engine rpm? Carry out the hypothesis test and include all the relevant steps (write out H0 and H1 , determine the appropriate test-statistic with or without using your calculator, determine the critical test-statistic or p-value using your calculator, and come to a conclusion). Be sure to clearly include all relevant data points such as ↵, tcalculated , and p value. Page 7 ECON 2142 Graded Homework 2 8. (5 points) Given the following data from three independent samples, use the 0.025 level in determining whether the population means could be the same. The following multiple choice questions are geared towards evaluating this hypothesis. Sample 1 2 3 6 7 14 9 20 18 18 11 23 12 15 17 13 23 27 10 16 (a) (1 point) What is the formulation of H0 and H1 ? A. B. C. D. H0 H0 H0 H0 : µ1 : µ1 : µ1 : µ1 = µ2 = µ2 = µ2 = µ2 = µ3 = µ3 = µ3 = µ3 = µ4 = µ5 = µ6 versus H1 : at least one of the µ’s does not equal another = µ4 = µ5 = µ6 versus H1 : µ1 6= µ2 6= µ3 6= µ4 6= µ5 6= µ6 versus H1 : at least one of the µ’s does not equal another versus H1 : µ1 6= µ2 6= µ3 (b) (1 point) How many treatments are there and is this a balanced experiment, respectively? A. B. C. D. 3; 3; 6; 6; No Yes No Yes (c) (1 point) In the ANOVA test, how many degrees of freedom are associated with the treatments and error, respectively? A. B. C. D. 2; 3; 2; 3; 14 14 17 17 (d) (1 point) What is the value of Fcalculated ? A. B. C. D. 0.025 0.048 3.80 4.86 (e) (1 point) At the 0.025 level, what is the conclusion? A. B. C. D. We reject H0 and conclude that the means of the four di↵erent samples may be equal We fail to reject H0 and conclude that the means of the four di↵erent samples may be equal We reject H0 and conclude that the means of the four di↵erent samples may not be equal We fail to reject H0 and conclude that the means of the four di↵erent samples may not be equal Page 8 ECON 2142 Graded Homework 2 9. (6 points) Researchers have obtained and tested samples of four di↵erent brands of nylon rope that are advertised as having a breaking strength of 100 pounds. Given the breaking strengths (in pounds) shown below, use the 0.025 level in comparing the brands. Fill in the ANOVA table as part of your solution (note that not every cell is required to be populated; you should know which ones are necessary and which ones aren’t). Brand A 103.1 108.9 106.7 114.3 113.3 110.5 Brand B 111.6 117.8 109.8 110.1 118.3 116.7 Brand C 109.0 111.8 113.0 109.7 108.6 114.7 Brand C 118.0 115.8 114.2 117.3 113.8 110.6 ANOVA Table: Source of Variation Between Groups Within Groups Total SS df MS Conclusion: Page 9 Fcalculated p-value Fcritical ↵ ECON 2142 Graded Homework 2 10. (5 points) A national public television network has found that 35% of its contributions are for less than $20, with 45% for $20–$50, and 20% for more than $50. In a random sample of 200 of the contributions to a local station, 42% were for less than $20, 43% were for $20–$50, and 15% were over $50. At the 0.05 level, does the local station’s distribution of contributions di↵er significantly from that experienced nationally? The following multiple choice questions are geared towards evaluating this hypothesis. (a) (1 point) In this hypothesis test, the determination of 2critical relies on m = the number of parameters that had to be estimated in computing the expected frequencies. What is the value of m? A. B. C. D. 4 2 1 0 (b) (1 point) What is the sum of the di↵erences between the observed and expected frequencies? What is the sum of the squares of the di↵erences between observed and expected frequencies? A. B. C. D. 0; 0 0; 5.48 0; 312 5.48; 312 (c) (1 point) What is the conclusion of the hypothesis test? A. We reject H0 and conclude that the distribution of contribution amounts for this network is equal to the national level distribution. B. We reject H0 and conclude that the distribution of contribution amounts for this network is not equal to the national level distribution. C. We fail to reject H0 and conclude that the distribution of contribution amounts for this network is equal to the national level distribution. D. We fail to reject H0 and conclude that the distribution of contribution amounts for this network is not equal to the national level distribution. (d) (1 point) Is the A. B. C. D. statistic ever a negative value? It is possible under some circumstances for the 2 statistic to be negative Under no circumstances can the 2 statistic be negative The 2 statistic is always negative Cannot determine the determine the nature of the 2 statistic with limited information (e) (1 point) Is the A. B. C. D. 2 2 distribution symmetric and if so, is it positively or negatively skewed? symmetric; positive skew not symmetric; negative skew not symmetric; positive skew symmetric; no skew Page 10 ECON 2142 Graded Homework 2 11. (6 points) The personnel director of a large firm has summarized a random sample of last year’s absentee reports in the accompanying contingency table. At the 0.01 level of significance, test the independence of age versus length of absence. The following long answer questions are geared towards evaluating this hypothesis. Note: you can disregard the fact that some of the cells have frequency < 5. In class, we discussed this as being a requirement, but for the sake of this question, you can disregard this fact. Age Group 25 years 26-40 years 40 years Number of Days Absent 1 2-4 5-7 > 7 30 8 3 4 45 15 12 5 7 39 18 22 11 10 61 63 42 19 21 145 (a) (1.5 points) What are the null and alternative hypotheses? (b) (1.5 points) Populate the following table using the expected frequencies if the age and length of absence are independent. 1 Age Group Number of Days Absent 2-4 5-7 >7 25 years 26-40 years 40 years 63 (c) (1.5 points) What is the value of 42 19 2 calculated ? (d) (1.5 points) What is the conclusion at the 0.01 level of significance? Page 11 21 45 39 61 145 ECON 2142 Graded Homework 2 12. (5 points) Safety researchers in a government agency believe that too much variability in the speeds of vehicles on urban sections of interstate highways can contribute to accidents by causing a greater level of interaction between vehicles traveling in the same direction. They believe that a standard deviation in excess of 5 mph is undesirable. Observing a random sample of vehicles on an urban portion of the interstate highway in their locale, they find the speeds to be as listed below: 78.9 70.1 58.1 54.7 64.3 51.4 58.9 55.2 59.4 52.9 48.1 57.5 58.4 52.2 58.8 48.1 71.9 57.8 63.1 55.3 53.6 59.1 64.4 55.4 57.7 64.4 59.2 59.6 68.6 60.0 62.3 64.8 57.0 49.9 58.2 55.9 68.1 70.0 63.2 59.9 55.0 60.6 59.7 64.4 62.9 52.4 65.6 61.0 (a) (2.5 points) At the 0.025 level, and assuming a normal distribution of vehicle speeds, could they be mistaken in their conclusion that too much variability exists in the speeds of vehicles passing this location? (b) (2.5 points) Determine the 95% confidence interval for the population variance of the speeds of vehicles passing this location on the urban interstate. Page 12 61.7 62.1 61.7 ECON 2142 Graded Homework 2 13. (6 points) As of mid-June 2009, five U.S. Government Bond mutual funds were reported as having generated the 1-year and 3-year (annualized) rates of return shown below. U.S. Government Bond Fund American Funds, AMUSX Fidelity, FGOVX Wells Fargo, STVSX Morgan Stanley, USGBX MFS Govt. Securities, MFGSX x=3-Year Annualized Rate of Return 6.3% 7.4 6.5 2.4 7.2 y=1-Year Rate of Return 7.1% 8.4 7.3 1.5 9.0 (a) (2 points) Determine the least-squares regression line and interpret its slope. (b) (2 points) For a U.S. Government Bond mutual fund with a 3-year annualized rate of return of 5.0%, estimate the 1-year rate of return. (c) (2 points) For a U.S. Government Bond mutual fund with a 3-year annualized rate of return of 7.0%, estimate the 1-year rate of return. Page 13 ECON 2142 Graded Homework 2 14. (4 points) The following questions are two short answer questions related to the previous question and to the topic of simple linear regression overall. (a) (2 points) Using the previous problem, construct and interpret the 90% confidence and prediction intervals associated with a 3-year annualized rate of return of 7.0%. (b) (2 points) Describe the standard error of the estimate (SEE) and demonstrate how it relates to the sum of squared errors (SSE). Page 14 ECON 2142 Graded Homework 2 15. (5 points) A tire company has carried out tests in which rolling resistance (pounds) and inflation pressure (pounds per square inch, or psi) have been measured for psi values ranging from 20 to 45. The regression analysis is summarized in the following calculator output: (a) (1 point) To the greatest number of decimal places in the output, what is the least-squares regression line? (b) (1 point) What proportion of the variation in rolling resistance is explained by the regression line? (c) (1 point) At what level of significance does the slope of the line di↵er from zero? What type of test did the calculator use in reaching this conclusion? (d) (1 point) At what level of significance does the coefficient of correlation di↵er from zero? Compare this with the level found in part (c) and explain either why they are di↵erent or why they are the same. (e) (1 point) Construct the 95% confidence interval for the slope of the population regression line. Page 15 ECON 2142 Graded Homework 2 16. (5 points) For n = 5 data points, the following quantities have been calculated: P P x = 457.8 x2 = 44, 195.74 (a) (1 point) What is the value of b1 ? (a) (b) (c) (d) P P 2y = 1030.1 y = 215, 344.8 P P xy =2 96, 706.05 (y ŷ) = 617.5735 0.90 1.05 1.20 1.35 (b) (1 point) What is the value of b0 ? (a) (b) (c) (d) 90 100 110 120 (c) (1 point) What is the value of the SEE? (a) (b) (c) (d) 14.35 15.35 16.35 17.35 (d) (1 point) What is the coefficient of correlation, r? (a) (b) (c) (d) 0.50 0.70 0.90 1.10 (e) (1 point) What is the interpretation of r2 generally? (a) (b) (c) (d) The The The The percent percent percent percent of of of of variation which is explained by the regression correlation which is explained by the regression variation which is NOT explained by the regression correlation which is NOT explained by the regression Page 16 ECON 2142 Graded Homework 2 17. (15 points) Annual per-capita consumption of all fresh fruits versus that of apples and grapes from 1998 through 2003 was as shown in the table below: Year 1998 1999 2000 2001 2002 2003 y =All Fresh Fruits 128.9 lb/person 129.8 128.0 125.7 126.9 126.7 x1 =Apples 19.0 lb/person 18.7 17.6 15.8 16.2 16.7 x2 = Grapes 7.1 lb/person 8.1 7.4 7.7 8.7 7.5 We get the following regression output from the calculator: (a) (2 points) Interpret the partial regression coefficients. (b) (2 points) What is the estimated per-capita consumption of all fresh fruits during a year when 17 pounds of apples and 6 pounds of grapes are consumed per person? Page 17 ECON 2142 Graded Homework 2 (c) (2 points) Determine the 95% prediction interval for per-capita consumption of fresh fruits during a year like the one described in part (b). (d) (2 points) Determine the 95% confidence interval for mean per-capit ...
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