Statistics 252 - Homework # 2 - Winter 2016 STAT 252 Homework 2 (44 marks) - Due Thursday, February 25 by 10 pm Show all your work and include appropriate justification to receive full credit for a solution. Single numerical answers (unless indicated) will not receive any marks. In particular: Whenever you are asked to \"carry out the most appropriate test\": i) Define the parameter(s) of interest. ii) State clearly the null and alternative hypothesis in terms of the parameter(s). iii) Calculate the test statistic and state all of its components. T-test: State the estimate, hypothesized value and standard error. F-test: State the sum-of-squared residuals and degrees of freedom for each model. iv) Calculate the p-value for the test. Give a probability statement identifying how the p-value is calculated. That is, show how the test statistic is compared to the null distribution of the test statistic. If the p-value is given in output, then it is fine to report it as is. If it is not, then you must estimate the p-value (within a range of values), using the appropriate table. v) Answer the question in plain English by referring to the p-value (and a significance level if given). Whenever you are asked for a \"confidence interval\": i) ii) iii) iv) v) State the estimate. State the standard error. State the critical value. Calculate the interval. Interpret the interval. NOTE: If the exact distribution you need is NOT on the corresponding table, round your degrees of freedom DOWN to the nearest integer. There are no \"extensive\" computations required for questions 1 and 2. 1. (2 marks) A psychology student, interested in the migratory eating habits of human organisms during the lunch hour on the university campus, breaks up \"Quad\" into six regions. Over the course of two full weeks, the student measured the organism concentration for each day in each region. Set up the null and alternative hypotheses to test for any difference between the average concentrations of humans among the regions. What is the distribution of the test statistic under the null hypothesis? 2. (2 marks) Consider a situation where we have eight sections of Statistics 252. There are four instructors and each instructor teaches two sections. Each section has 50 students enrolled. Define the null and alternative hypotheses for the test to determine if any instructor has different marks between their own sections. What is the distribution of the test statistic under the null hypothesis? 3. (7 marks in total) A study was designed to evaluate if varsity athletes of certain heights tended to participate in particular sports. At a local university, three sports (involving both genders) were used and athletes were randomly selected. The table below summarizes the heights of athletes (in inches) from each team in the study. The summary statistics and ANOVA table for these data are given on the following page. Assume all required assumptions are satisfied. a) (6 marks) Carry out the most appropriate test to determine if there are any differences in height among the six teams. b) (1 mark) What is the best estimate for the common (pooled) standard deviation of the 6 groups? (Hint: this value can be found (with minor calculation) from a value in the completed ANOVA table.) 1 Statistics 252 - Homework # 2 - Winter 2016 Group 1 2 3 4 5 6 Team Men's volleyball Women's volleyball Men's basketball Women's basketball Men's hockey Women's hockey ni 17 18 14 15 26 25 Sample Mean 76.29 71.11 76.36 70.93 70.92 66.60 Sample S.D. 3.197 2.447 4.272 2.282 3.084 2.449 Height Sum of Squares Between Groups Within Groups Total df Mean Square 267.469 F Sig. 957.301 114 4. (15 marks in total) A nostalgic hockey fan wanted to compare the 4 teams that became part of the NHL in 1979. The fan measured the winning percentage (in decimal form) for eight randomly selected seasons for each team. Use the tables below to help answer the questions. Assume all required assumptions are satisfied. ANOVA Table Source of Variation Between (Treatment) Within (Error) Total Group 1 2 3 4 Sum of Squares 0.3035 0.3613 0.6647 Team Edmonton Oilers Hartford Whalers Qubec Nordiques Winnipeg Jets Conference Western Eastern Eastern Western df 3 28 31 Mean Square 0.1012 0.0129 Country Canada U.S.A. Canada Canada Mean 0.6499 0.4092 0.4894 0.4135 F-Statistic 7.84 p-value 0.001 St. Dev. 0.0944 0.0664 0.1625 0.1090 a) (2 marks) Define a linear combination of the four population means that compares the overall average winning percentage for the Western teams to the overall average winning percentage for the Eastern Conference teams (mean for Western - mean for Eastern). b) (3 marks) Calculate a 95% confidence interval for the linear combination in part (a). Based on the interval, do the Western conference teams have a different overall winning percentage than the Eastern teams? Why or why not? c) (5 marks) Repeat parts (a) and (b) by creating a linear combination of the four means to compare Canadian and U.S. teams (mean for Canadians - mean for U.S.). Then, based on the interval, do Canadian teams have a different overall winning percentage than the American teams? Why or why not? d) (5 marks) Using the Bonferroni method for simultaneous inference, summarize all differences using a family-wise confidence level of 94%. Note: You do not have to construct all the confidence intervals. You can simply find ME, the margin of error (it will be the same for all pairs as we have equal sample sizes), and then find all pairs of sample means that differ by at least ME. Clearly show your calculation for the ME. 2 Statistics 252 - Homework # 2 - Winter 2016 5. (8 marks in total) Consider a problem comparing I = 17 means. Three models are to be considered; the onemean model, a J-mean model, and the 17-mean model. Fifty observations were collected for each group. Use the given information in the tables below to help answer the questions. Assume all required assumptions are satisfied. a) (2 marks) Fill in the tables above. What is J? b) (6 marks) Carry out the most appropriate test to see if the 17-mean model is better than the J-mean model. Table 1: ANOVA table for testing the 17-mean model vs. the one-mean model: Source of Variation Sum of Squares df Mean Square Between (Extra) 16 Within (Full) 833 20 Total (Reduced) 64008 Table 2: ANOVA table for testing the J-mean model vs. the one-mean model: Source of Variation Sum of Squares df Mean Square Between (Extra) 36018 Within (Full) 33.32143 Total (Reduced) F-Statistic 147.96 p-value F-Statistic p-value <0.001 6. (10 marks in total) To assess the effects of three different gasoline supplements on average mileage (mi/gal), researchers conducted an experiment using 60 automobiles of the same type, model, and engine size, with 10 automobiles randomly assigned to each gasoline blend. The six gasoline blends and the resulting data appear in the accompanying table. Group 1 2 3 4 5 6 Description Control Control + Supplement A Control + Supplement B Control + Supplement C Control + Supplement A + Supplement B Control + Supplement A + Supplement C ni 10 10 10 10 10 10 Sample Mean 25.5999 26.7061 28.2070 33.2476 34.1239 37.3222 Sample S.D. 5.59255 5.13037 4.00454 4.50070 5.80158 4.88503 Note: Because of the chemical makeup of supplement B and C, they could not be combined in the same blend. a) (5 marks) Suppose you want to determine if there is an overall supplement A effect. Define a linear combination for this effect and calculate a 98% confidence interval for the linear combination. b) (5 marks) Repeat the process in part (a) for the overall supplement C effect. 3