(e) ( 5 points) Suppose the total number of seats in the auditorium room is 200. We can define the occupancy as the number of audience members divided by 200. Now we want to conduct a statistical test to know whether the mean occupancy is lower than 0.7. Would you suggest to use the t-test for the mean or the z-test for the proportion? Give your reason according to the assumption that you can make on the randomness of the data. (f) ( 5 points) If we assume the population variance is 252, please evaluate the type ll error rate of your test in (b) when the true population mean is 145. (g) ( 5 points) Similar to question (f), we can evaluate the detection power under different assumptions of the true population mean. Which one of the zo.25 A following plots would be the most likely power curve for this design? (A) B (C) 140 /45 0.0 0.2 0.4 0.6 0.8 1.0 power power power 00 0.2 0.4 0.6 0.8 00 02 0.4 0.5 0.8 120 130 140 150 150 120 130 140 150 190 120 130 140 150 160 u (h) ( 5 points) If we increase the number of samples, what would the power curve look like? Please draw two curves with sample sizes 50 and 100.2. The following data represent the number of audience members per week at a theater. The data was randomly retrieved from the past records. 163 165 94 137 123 95 170 96 117 129 152 138 147 119 166 125 148 180 152 149 14 = 136. 22 167 120 129 159 150 119 113 147 169 151 116 150 110 110 143 90 134 145 156 165 1 = 24 46 174 133 128 100 86 148 139 150 145 100 The average is 136.22 and the standard deviation is 24.44. (a) ( 5 points) Please estimate the mean number of audience members attending the activity with a 95% confidence interval. (b) (10 points) According to the data, can we say that the mean number of audience per week is significantly different from 140? Please conduct the test with significance level 0.05 using the rejection region method. (c) ( 5 points) Questions (a) and (b) are answered based on the normal assumption of the data. How would you evaluate the normality of the data? Please give you suggestions. You do not need to do it. (d) ( 5 points) If the normality assumption of the data is not valid, can we still do the same analysis for (a) and (b)? Why or why not