Fall 2016 STAT 200 Written Assignment #2 (Total mark: 50) Due: Friday November 18, 2016 Instructions: Typewritten answers are preferred. If your handwriting is illegible or if the answers are not presented in a neat and organized manner, with answers clearly numbered and blank lines separating answers, you may lose 2 marks. Please leave one inch margins for marking. If you use R Commander to compute probabilities for any of the questions, please state clearly you have done so and provide parameter values that you input in R Commander. 1. The amount of time spent by students who visit the library on Saturdays is a random variable with mean 3 hrs and standard deviation 0.9 hrs. Fifty percent of students who visit the library on Saturdays spend more than 3.2 hrs at the library. (a) Explain why the amount of time spent by students who visit the library on Saturdays does not follow a Normal model. [2 marks] (b) Consider the following two scenarios: (i) Ten students who visit the library on Saturdays are to be randomly selected and their average time spent at the library is then computed. (ii) Fifty students who visit the library on Saturdays are to be randomly selected and their average time spent at the library is then computed. Describe the distribution that the average time spent follows in each scenario. Specify the mean, standard deviation and shape for each of the two distributions. Justify your answers and make sure to state all assumptions. If you cannot come up with an answer, state so and briefly explain why not. [8 marks] (c) A librarian now takes 30 random samples of 15 students and calculates the average time spent by the 15 students in each of the 30 random samples. He also takes 10 random samples of 40 students and caluclates the average time spent by the 40 students in each of the 10 random samples. Which of these two scenarios do you expect to see a smaller variability in the average time spent at the library on Saturdays : the 30 random samples of 15 students or the 10 random samples of 40 students? Justify your choice and make sure to state all assumptions. [4 marks] (d) A new librarian is hired and he randomly samples 100 students who visit the library on Saturdays. From his sample, 33 students spend more than 3.2 hrs at the library. Is the number of students who spend more than 3.2 hrs at the library in the sample unsually high, low, or neither? Justify your answer using probability and make sure to state all assumptions. [6 marks] 2. According to a German candy factory, 98% of gummy bears that are produced in this factory weigh at least 2.2 grams. One customer wanted to test the factory's claim on the true proportion of gummy bears that weigh at least 2.2 grams. He decided to buy a bag of gummy bears from this factory and weigh each piece. In the bag that he obtained, there were a total of 650 pieces and 520 weighed at least 2.2 grams. (a) Use the customer's data to construct a 99% confidence interval for the true proportion of gummy bears that weigh at least 2.2 grams. Interpret the confidence interval in the context of the question. [5 marks] STAT 200 Assignment 2 (continued) (b) Based on the confidence interval obtained in part (a), do you think 98% is a plausibe value for the true proportion of gummy bears that weigh at least 2.2 grams? Justify your answer. [2 marks] (c) Moreover, the factory claims that the number of pieces of gummy bears varies from bag to bag, but the bags have a mean of 650 pieces and a standard deviation of 15 pieces. To check this claim on the true mean number of pieces per bag, the quality control division of the factory obtained a random sample of 36 bags of their gummy bears and counted the number of pieces in each bag. The average number of pieces in the 36 bags was found to be 648. Do you think the result obtained by the quality control division is compatible with the factory's claim? Justify your answer using probability and make sure to state all assumptions. [6 marks] 3. UBC Thunderbirds won its fourth Vanier Cup last year, and some students suggest that we should expand our football stadium. Before spending millions of dollars, the school would like to ask for students' opinion. If more than two thirds of all UBC students agree, this plan will be proposed to the Board of Governors. In order to understand student's opinion, the Athletic Department conducted a survey on a random sample of students. From the 64 responses they collected, 47 agree with the expansion plan. The department will conduct a hypothesis test at the 1% significance level to determine if there is sufficient evidence to support submitting a proposal to the Board of Governors. (a) What is the parameter of interest to the Athletic Department? [1 mark] (b) What are the null and alternative hypotheses? Define any notation that you use. [3 marks] (c) Compute the test statistic. Show your work. [2 marks] (d) Compute the P-value. Show your work. [2 marks] (e) Is there strong evidence to convince the Athletic Department to make the proposal? Justify your answer. [3 marks] (f) What is the Type II error in this hypothesis test (in the context of the question)? Keep your answer to no more than 30 words. [2 marks] (g) Suppose the Athletic Department wants to re-conduct the survey and wish the margin of error for the 95% confidence interval for the parameter identified in part (a) to be no larger than 0.08. What is the minimum number of students they should ask? [4 marks] Page 2