final 2009 - Student Name: Please Print Student Number: M...

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Unformatted text preview: Student Name: Please Print Student Number: M Section: 001 002 003 Instructor: M. Yalovsky P. D’Antoni D. Hart M/W 1:05 —2:25 T/Th 2:35-3:55 M/W 10:05-11:25 Section: 004 005 006 Instructor: D. Hart D. Zhang D. Zhang M/W 2:35—3:55 T/Th 8:35-9:55 T/Th 10:05-11:25 McGill University Desautels Faculty of Management FINAL EXAMINATION December 9th, 2009 2:00PM —— 5:00 PM Business Statistics MGCR 271 Examiners: P. D’Antoni, D. Hart, M. Yalovsky, D. Zhang INSTRUCTIONS: MM 1. Please write your Name and Student Number on the exam paper. Circle the section Number in which you are registered. This is a closed book examination. Calculators that store text will not be permitted. Translation Dictionaries are permitted NQIAPP’NT‘ Note: Each student is permitted to bring with them a crib sheet (81/2xl l-two sided, either written or printed) r—Ar—Aom Ho. . Statistical tables are provided. The allotted marks for each question are indicated to the left of the question. All examinations, tables and crib sheets must be handed in upon completion of the exam. Total Pages for this examination: 17 (including the cover sheet) Please answer each question in the space allotted following the question. Use the back of the pages for rough work. Best of Success (7) Question 1 The new Director of the Human Resources Department of a large firm wishes to estimate the average number of working days lost to absenteeism per year for its group of professional employees. She has been told that in similar studies carried on in the past the standard deviations this firm follows a Normal distribution. a) Determine the number of professional employee’s files that should be sampled in order to obtain a 96% confidence interval for the mean which will have a margin of error of no more than two days. ' b) Suppose the sample size determined in part a) above had a mean of 6.30 days and a standard deviation of 4.57 days. Determine the 95% confidence interval for the mean. (8) Question 2 Jupiter Media used a survey to determine how people spend their free time. Watching television was the most popular activity selected by both men and women. In a sample of 800 men, 248 selected watching television as their most popular leisure time activity. In a sample of 600 women, 156 selected watching television as their most popular leisure time activity. a) Establish a hypothesis that can be used to test for a difference between the proportion of men and the proportion of women who selected watching television as their most popular leisure time activity. Conduct the hypothesis test and compute the p-value. State your conclusion at 5% level of significance. b) Determine a 95% confidence interval for the difference between the population proportions. (3) Question 3 Containers of coffee on a production line are to be filled with 16 oz of coffee..The quality control staff of the company has set up a system to monitor the mean content of all cans that are filled on this production line. Based on past experience it is known that the amount of coffee per can filled on the production line follows a Normal distribution with standard deviation 0.1 02. Every hour a sample of 9 cans is selected and their mean content is measured. Overfill and under-fill of cans is problematic. ' a) b) You have been asked to establish a hypothesis testing procedure by which you can perform automated hourly tests to determine if the machine performance is satisfactory. Your test will be based on the mean of these 9 hourly observations and you are asked to employ a 5% level of significance for each test. Clearly outline the procedure that you would develop based on the averages of each hourly sample of 9 independent cans. If the true mean content during a particular period has shified to 16.1 oz, determine the probability that the test developed in a) will correctly detect this deviation from the target value of 16 oz. Question 4 (7) Consider a random variable X that takes on one of two values 0 and 1 such that P(X=1) = p and P(X=0) = 1—p. Suppose we wish to test the hypothesis Hoip:% Ha:p=% We take two independent observations X1 and X 2 of this random variable and set the following decision rule. Do not reject Ho ifeither X1 + X2 = 0 g + X2 = 1, otherwise reject Ho. a) Determine the probability of meg] error b) Determine the power of the test. (6) Question 5 People at risk of sudden cardiac death can often be identified through the change in a signal averaged electro cardiogram before and after having undertaken prescribed activities. The current method employed is about 80% accurate. Hoping to improve the accuracy of prediction, a new testing procedure has recently been developed. The new method is tested on 50 people and correct results were obtained for 46 of the 50 patients. You have been asked to test whether these results provide convincing evidence that the new method is more accurate. Establish the appropriate hypothesis and carry out the statistical test. Your conclusion should provide information on the P-value of the test. Question 6 An undergraduate student wishes to compare the cost of one and two bedroom apartments in the area of the McGill campus. She collects data for a simple random sample of 10 advertisements of each type and records the advertised rent. She knows that given tight rental market in the area, renters typically pay the advertised price to secure the apartments. Historically, rents for one and two bedroom apartments in the area have been approximately normally distributed. Listed below are the rents advertised for one and two bedroom apartments (in $ per month). (10) Some summary statistics are given below) _ 611507 531.50 Standard deviation 88.26 83.33 a) Determine a 95% confidence interval for the additional cost of a second bedroom. b) Having takena course in Statistics, the student wishes to determine if two bedroom apartments rent for significantly more than one bedroom apartments. Carry out the appropriate test of hypothesis; provide your conclusion and the p—value associated with the test. c) Can you conclude that every one bedroom apartment costs less than every two bedroom apartment? d) Which do you think is the more useful method of analysis for a student planning to rent an apartment in the University area —-— the confidence interval or a test of significance? Justify your answer. (Additional working space is also provided on the next page.) Question 7 (7) a) Is there evidence of a significant difference amon shopping day? Conduct an appropriate hypothesi b) What factors not given might be relevant to this study? g the age groups with respect to major s test to answer the question. Use a=5%. Question 8 (13) The McGill Health Club has decided to initiate a drive to recruit new members. The managers believe that there might exist a relationship between the amount of advertising spent during a one month period and the number of new members recruited during that one month period. The data below presents a partial history of data gathered over the last 20 months. Advertisin Amount in $ ——_ __— _—_ To what extent can the number of newly recruited members be predicted by the amount of dollars spent on advertising? We have carried out some statistical analysis of the data and the partial results are as follows: ' ' 1000 Advertising $ Regression Analysis: New Members versus Advertising 3 The regression equation is New Members = Predictor Coef SE Coef T P Constant 44.597 3.064 Advertising 3 0.010580 0.003607 S = R-Sq = 32.3% R—Sq(adj) = 28.6% Analysis of Variance Source DF SS MS F P Regression 1 647.67 647.67 8.60 Residual Error 18 1355.28 Total 19 2002.95 ‘ Residual Plots for New Menfliers Normal Probability Plo‘tf__ I I r , VerSus Fits 99 g , V , 3 Q , “ 90 M mgr. I I 20 z ; ,r -— . z x 50 E; ' 3% 10 E : ' § 3 0 1'“ “ P , , r g ; r , , -10 ' -20 -10 ,0 10 I 20- 45 50 / 55 .60 65 Residual Fitted Value, “ r ' Versus Order 8 1012 14 16 18-20 Observation Order IO Summarize the relationships. Might there be some problems with the data? Determine the equation of the least squares regression line of new members on dollars expended on advertising. Insert the estimated regression line on the plot. Would you conclude that there is a significant linear relationship between the number of new members and the amount spent on advertising? Determine a 95% CI for the slope. If your advertising budget had $500 more how many additional new members would you expect that the Health Club could recruit? Provide a 95% confidence interval for your estimate. 7 Determine the coefficient of determination and explain its significance. ll (12) 1 Question 9 Alumni donations are an important source of revenue for colleges and universities. If administrators could determine the factors that could lead to increases in the percentage of alumni who make donations, they might be able to implement policies that could lead to increased revenues. Research shows that students who are more satisfied with their contact with teachers are more likely to graduate. As a result, one might suspect that smaller class sizes and lower student—faculty ratios (SFRatio) might lead to a higher percentage of satisfied graduates, which in turn might lead to increases in the percentage of alumni who make a donation. A regression analysis is done to investigate this issue. The following multiple regression model is used: GivingRate = [30 + [31 (Public) + [32 (GradRate) + [33 (ClassSize) + [34(SFRatio) + .04, with the usual assumptions on the error term a. In‘the above model, the “GivingRate” variable represents the percentage of alumni who made a donation to the university; the “Public” variable takes on the value 1 for public universities and 0 for private universities; the “GradRate” variable indicates the percentage of students who initially enrolled at the university and graduated; the ‘ “ClassSize” variable shows the percentage of classes offered with fewer than 20 students; the “SFRatio” variable denotes the student /faculty ratio or the number of students enrolled divided by the total number of faculty in the University. A partial statistical output is given below: SUMMARY OUTPUT Regression Statistics Multiple R 0816431673 R Square 0.666560677 Adjusted R Square 0.610987457 Standard Error 8392626673 Observations 29 ANOVA ’ df SS MS F Significance F M Regressmn Residual Total —————————____________________________________ ——-———————————————_—_______—_—___ Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -51.27966052 27.74872385 ~108.5502114 5.99089032 Public 1.284410374 3.448307546 -5.83254656 8.4013673] GradRate 0.985063232 0.23661 1042 0.496722047 1.47340442 ClassSize 0067798364 021310043 0372019304 050761603 SFRatio 042158072 068627319] -1 .837978961 0.99481752 M 12 (a) Write down the equation of the estimated regression model. (b) A private university has the following data values: “GradRate” = 92, “ClassSize” = 69, “SF Ratio” = 7. Predict the “GivingRate” based on the regression model. (c) Should “SFRatio” be included in the model given the other variables are included in the model? Provide a statistical justification for youranswer. Use a=5%. (d) Test for overall significance of the regression model. Clearly state the hypothesis and conclusion. (e) A second analysis using “SFRatio” as the only explanatory variable is conducted. A partial output from the statistical analysis is reported below. For the sake of parsimony you wish your model to have as few explanatory variables as possible. Can the model containing the “SFRatio” as the only explanatory variable predict the “GivingRate”? Fully explain. SUMMARY OUTPUT Regression Statistics Multiple R 0644029506 R Square 0.414774004 Adjusted R Square 0.393098967 Standard Error 10.48274547 Observations 29 ANOVA df SS MS F Signzficarzce F W Regression Residual Total 5069.793103 ———————________________ M Coeflicients Standard Error t Stat P—value Lower 95 % Upper 95% Intercept 51.18915192 5.161866032 9.916792 1.698E-10 40.59787779 61.78042604 SFRatio -l.87186854 0.42790711 -4.374474 0.0001633 -2.749861392 —0.99387568 l3 Question 10 For each of the following situations, indicate whether the statement is correct or incorrect. Justify your response. (10) a) b) C) d) e) A researcher wishing to test equality of two population’s means formulates the hypothesis as H0 : Y1 = Y2 versus two sided alternative hypothesis Ha : Y1 7E Y2 . A two sample 2 test statistic for testing the difference in proportions gave a P—value of 0.96. From this you can reject the null hypothesis of equality of proportions with 95% confidence. If the P-value for a test of significance is 0.35, we can conclude that the null hypothesis has a 35% chance of being true. In a multiple regression with a sample size of 50 and 4 explanatory variables, the test of the null hypothesis H0 :32 = 0 uses a t test statistic that follows a t- distribution with 45 degrees of freedom under the assumption that the null hypothesis is true. One of the assumptions for multiple regressions is that each of the explanatory variables follows a normal distribution. - 14 15 (12) Question 11 E] Iron-deficiency anemia is the most common form of malnutrition in developing countries, affecting about 50% of children and women and 25% of men. Iron pots for cooking foods had traditionally been used in many of these countries, but they have been largely replaced by aluminum pots, which are cheaper and lighter. Some research has suggested that food cooked in iron pots will contain more iron than food cooked in other types of pots. One study designed to investigate this issue compared the iron content of some Ethiopian foods cooked in aluminum, clay, and iron pots. One of the foods was yesiga wet’, beef cut into small pieces and prepared with several Ethiopian spices. The iron content of four samples of yesiga wet’ cooked in each of the three types of pots is given below. The units are milligrams of iron per 100 grams of cooked food. , 1A partial computer output is given below: ANOVA: Single Factor SUMMARY Groups Count Sum Average Variance Aluminum 4 8.23, 2.0575 0.241492 Clay 4 8.71 2.1775 0.386025 Iron ' 4 18.72 4.68 0.394733 ANOVA Source 0 Variation SS d MS F P-value F crit Between Groups Within Groups Total 20.606 (a) Based on the above data test whether the average iron content is the same for all three types of pots at 5% level of significance. Carry out the analysis of variance. Report the F statistic with its degrees of freedom and P-value. State your conclusion. (b) What assumptions are made in order to carry out the analysis of variance? (0) Conduct a multiple comparisons procedure to determine which pairs of means differ significantly. Summarize you results. 16 l7 ...
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final 2009 - Student Name: Please Print Student Number: M...

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