Mid-Term_-_Solutions - UNIVERSITY OF TORONTO QUANTITATIVE...

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Unformatted text preview: UNIVERSITY OF TORONTO QUANTITATIVE METHODS IN ECONOMICS II ECMB12H3F (LEC02) MID-TERM TEST – SOLUTIONS DURATION: 90 MINUTES OCTOBER 20, 2010 IMPORTANT: (i) This test should be answered in BALLPOINT PEN and students who attempt the test in pencil would surrender their rights to request for re-assessment. (ii) Ensure HANDWRITING is LEGIBLE and show STEP-BY-STEP CALCULATIONS. Answer all questions only in the DESIGNATED PAGES. ROUGH WORK should be done in the LAST TWO PAGES and contents in those pages will not be graded. (iii) AIDS ALLOWED – a HAND-WRITTEN SINGLE-SIDED LETTER SIZE CRIB SHEET (photocopies or printed copies not allowed) and a NON-PROGRAMMABLE CALCULATOR (other electronic devices with calculator functions not allowed). (iv) DO NOT SEPARATE any page from this test. Student Name Student ID# Question 1 2 3 4 5 Total Marks 17 31 20 12 20 100 Score Page 1 of 8 1. As the vice president in operations of a major fast food chain, the current average time to serve a customer using the drive-thru service is 3 minutes and you are interested in cutting down the average time by installing a new automated self-serve system, which costs millions of dollars. The new system has been tested in a particular location where it served a sample of 64 customers and the average time is 2.75 minutes with a standard deviation of 1 minute. (i) Follow the six-step critical value approach to hypothesis testing at 5% level of significance with the sample data to determine whether to invest in this new automated self-serve system or not. (ii) Can you determine the range of the -value of the test statistic? (i) 12 marks and (ii) 5 marks (i) Step 1 – the company will only invest in the new automated self -serve system if it can significantly reduce the average time to serve customers using the drive-thru service, which implies a one tail test. and Step 2 – the level of significance is 5% and the sample size . Step 3 – only the -test is appropriate where there is no information about the population variance. Step 4 – for and for the one tail test is Step 5 – with sample statistics with degrees of freedom , the critical value . , , and . The test statistic Step 6 – since and it is less than the critical value of , the test statistic falls in the rejection region and hence the null hypothesis is rejected. There is enough evidence to conclude that the average time to serve a drive-thru customer is significantly less than 3 minutes and hence it is worthwhile to invest in the new automated self-serve system. (ii) With the degrees of freedom at 63, the probability of a value at or smaller is 2.5% and the probability of a value at or smaller is 1%. Since the test statistic of is between and , which implies the range of the -value of the test statistic of must be between 1% and 2.5%. Page 2 of 8 2. (i) Explain when a researcher may commit a Type I Error and how the probability of Type I Error is being determined. (ii) Suppose the population variance is known and the hypothesized population mean is false when the true population mean is in which . Use two distribution graphs (with one above the other and each graph has two value lines – one for units and one for sample unit of measurement for – as shown in lecture discussion) to help you explain the relationship between Type I and Type II Errors with and at . (iii) Use two separate distribution graphs (with the same set up as in part (ii)) to help you explain why the one tail test is more powerful than the two tail test . Any intuitive explanation? (i) 3 marks; (ii) graphs 4 marks and explanation 8 marks; and (iii) graphs 4 marks and explanation 12 marks (i) A researcher commits a Type I Error if he rejects the null hypothesis when it is true. The probability of Type I Error is , which is called the level of significance of the test, and is set by the researcher in advance. (ii) Given the hypothesized population mean is false in a lower tail test, any sample with a test statistic higher than when (defined as the Type I error) would not reject the null hypothesis. When the true population mean is lower than , the probability of that occurring is defined as the Type II error (indicated in the diagram as ). The lower the probability of Type I Error, the smaller the critical value, the more the red line moves to the left and the higher the probability of Type II Error under the true distribution in the lower graph. In other words, smaller Type I Error results in larger Type II Error. Page 3 of 8 (iii) With and , any sample with a test statistic higher than would not reject the null hypothesis even though it is false when the true population mean is lower than , and the probability of the Type II error for this lower tail test is the area under the true distribution in the lower graph (as pointed out in part (ii)). With and , any sample with a test statistic between and would not reject the null hypothesis even though it is false when the true population mean is lower than , and the probability of the Type II error for this two tail test is the area under the true distribution in the lower graph . Since , . The power of the test for the one tail test is and for the two tail test. Therefore, for and the one tail test is more powerful than the two tail test. A two tail test only aims to test if there is any difference from the hypothesized population parameter in either direction (larger or smaller) when the researcher does not have enough information to determine the direction of the test. A one tail test has specific direction of the difference because the researcher has more information in which more information often leads to better decision. Hence, the one tail test is more powerful than the two test tail. Page 4 of 8 3. Suppose you are working on a PhD dissertation regarding customer satisfaction of new car buyers between domestic brands and imported brands. You randomly selected a sample of 61 new car buyers of domestic brands (with sample average of 2.8 minor repairs and sample standard deviation 1.7 during the first three months) and 41 new car buyers of imported brands (with sample average of 2.4 minor repairs and sample standard deviation 1.2 during the first three months). Explain your decision on the selection of appropriate tests (all at level of significance) in each step to examine if the average number of minor repairs of new domestic cars is different from that of new imported cars. 20 marks Since there is no information on whether the population variances of the average minor repairs of these two groups of new cars are equal or not, an -test on the difference in population variances of these groups and must be performed before a decision can be made on whether to use the pooled- or separate-variance -test with unknown population variances. Given the sample variance of the minor repairs on new domestic cars is greater than that of new imported cars , the sample of new domestic cars is set as sample 1 and the sample of new imported cars is set as sample 2 . The critical value of this two tail -test at is 1.80. The test statistic level of significance with and Since is greater than the critical value, reject . There is enough evidence to conclude that the population variance of minor repairs on new domestic cars is different from that of new imported cars. Hence, the separate-variance -test for the difference between two means and is the appropriate test. The degrees of freedom must be calculated before we can determine the critical value: The standard error of the separate-variance: Page 5 of 8 The critical value for this two tail -test at 5% level of significance with degrees of freedom is . The test statistic Since is not in the rejection region, do not reject . There is not enough evidence that the average number of minor repairs of new domestic cars is different from that of new imported cars. Page 6 of 8 4. A survey done in 2006 indicated that 50% the females surveyed were impulsive shoppers when the economy was booming. The same marketing firm conducted the same survey earlier this year on a different random sample with 12 female respondents and found that 4 of them were impulsive shoppers. Use the -value approach to determine if the current recession has reduced the proportion of females as impulsive shoppers at 5% level of significance. (Hint: If you cannot find the exact test statistic in the -table, then use the approximation method discussed in class) 12 marks According to the 2006 survey, 50% of the females were impulsive shoppers . The first step is to check if the -test is appropriate. That is, if and . Using and , and . Passing this check, the -test is appropriate. It may look like there are two populations, one from 2006 and one from this year, but there is only one population so that there is no need to know the sample size of the survey done in 2006 because there is no need to check. To determine if recession has reduced the proportion of females as impulsive shoppers implies this will be a one tail test in the lower tail such that and From the sample taken earlier this year, and , . Since the of is not printed in the table, we must find the closet values from the table to approximate the -value of the . Given , if , then is definitely larger than and hence do not reject the null hypothesis. There is not enough evidence that the recent recession has reduced the proportion of females as impulsive shoppers. Page 7 of 8 5. A randomized block experiment has groups and blocks . (i) Complete the table below. (ii) What are the hypotheses in testing whether the randomized block design is advantageous to use and your conclusion at 5% level of significance? (iii) Compute the estimated relative efficiency and explain what it means. (i) 10 marks; (ii) 5 marks; and (iii) 5 marks Source Degrees of Freedom Sum of Squares Mean Square (Variance) Among Groups Among Blocks Error Total (ii) To test whether the randomized block design is advantageous to use, and Not all are equal (where With and , the critical value . Since , reject . There is enough evidence to conclude the randomized block design is advantageous to use. (iii) The estimated relative efficiency is defined as This value for relative efficiency means that it would take approximately 1.4 times as many observations in a one-way ANOVA design as compared to the randomized block design in order to have the same precision in comparing the groups. Page 8 of 8 ...
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This note was uploaded on 05/31/2011 for the course MGT C06 taught by Professor A.stawinoga during the Fall '10 term at University of Toronto- Toronto.

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