x + Z Question Four Let X1, X2, Xn be a random sample from a population that is normally distributed with mean д and variance o2. Is the sample variance defined by S=(X-X)2 unbiased for 2? Justify your answer. Question Five [7 marks] Derive the formula you would use to determine a symmetrical 100(1 - a)% confidence interval for the population mean of a normal population with unknown variance, based on a random sample of size n. Question Six [7 marks] Assume the population of battery lives to be normally distributed with mean and variance o2. If 5 batteries have lifetimes of 1.9, 2.4, 3:0, 3.5 and 4.2 years, construct a 99% confidence interval for μ. [8 marks] Question Seven In a sample survey, 140 of 500 people interviewed in a large city said they shop in a certain shopping mall at least once a week. Construct a 99% confidence interval for the true proportion of people in the city who shop in this shopping mall at least once a week. n=500 [7 marks] CL+I P(x) = (~) px M= np 40400 qn-x M= 2 x-2 1/1 ≤ μ x +2√ ((x) + u = 3 AL=CL+1 2