Question

Exercise 9. (10 points] Confidence Intervals A team of 200 reserachers studying tax record each receives a random sample of 500 taxpayers records. The project requires to compute a confidence interval for the average taxable income: the project guidelines suggest using the following estimator: Ŝ Ŝ c1= [X- X – 2.576 · X + 2.576 n -15 n-1 where § 21–1(Xi – X)2 is the sample standard deviation and X := 2 Xi is the sample mean. a) For example, researcher number #74 computed X = $51, 463.39 and ŝ = $12, 346.71. What confidence interval for the population mean did researcher number #74 obtain?

a) For example, researcher number #74 computed X¯ = $51, 463.39 and ˆs = $12, 346.71. What confidence interval for the population mean did researcher number #74 obtain?

b) The IRS knows that the true average taxable income is $ 52,345.23 Approximately, how many of the researchers will construct confidence intervals that do in fact contain $ 52.345.23? Justify your answer using a statistical result we have seen in class.

Answer 1

a) computed CI: (51463.39 = 2.576 * 12346.71/500) so ans : (50041.02,52885.76) 6) Note above CI uses Zer = 2.576, which means this is 99% con fidence interval and hence, out of 100 CIS, 99 will have true mean inside them so answer = 0.99 * 200 = 198 so about 198 will construct CIs that do in fact contain the true avg.