Discussion Questions 1 Let X,, X, and X, be independent, identically distributed random variables with mean u and variance of, and A = (X, + X, + X;)/3. Is A an unbiased estimator of ! 2 Suppose that A is a binomial random variable with parameters n and p. Is p = X an unbiased estimator of p? 3 From a random sample of 50 graduating students at a private college, the mean COPA is 2.95 with a standard deviation of 3.25. Construct a 90% confidence interval for the mean CGPA for all graduating students at this college. 4 Simple random samples of students are selected from two different faculties; 12 students from Faculty of Science and 16 students from Faculty of Business. The sample from Faculty of Science has an average CGPA of 2.75 with a standard deviation of 2.85, and the sample from Faculty of Business has an average CGPA of 2.95 with a standard deviation of 4.25. Determine the 95% confidence interval for the difference in CGPAs at the two faculties assuming that CGPAs in both faculties came from normal distributions. 5 In a study on household income in a small town, 64 families were randomly selected. It is found that 24 families have monthly household incomes of less than P2,500. Find a 90% confidence interval for the percentage of all families in the town with household incomes less than P2,500 per month. 6 Suppose two groups of adults were selected randomly and their heights were measured. The first group of 35 adults has mean height of 167 cm with a sample variance of 28.5 cm, whereas the second group of 30 has mean height of 164 cm with a sample variance of 32.4 cm. Find a 90% confidence interval for the true ratio of the two variances. A car manufacturer claimed that the average gas mileage of its new brand of hybrid car is 25 miles per liter. The gas mileages for ten randomly selected hybrid cars of the new brand are recorded as 24.4, 25.1, 22.6, 26.2, 25.3, 23.2, 21.9, 23.8, 24.5 and 25.0. Assume that the gas mileage is normally distributed. Does the sample data support the manufacturer's claim at the 0.01 significance level