6.15 Confidence interval mistakes and misunderstandings. Suppose that 500 randomly selected alumni of the University of Okoboji were asked to rate the university’s academic advising services on a 1 to 10 scale. The sample mean was found to be 8.6. Assume that the population standard deviation is known to be σ = 2.2.
(a) Ima Bitlost computes the 95% confidence interval for the average satisfaction score as 8.6 ± 1.96(2.2). What is her mistake?
(b) After correcting her mistake in part (a), she states, “I am 95% confident that the sample mean falls between 8.4 and 8.8.” What is wrong with this statement?
(c) She quickly realizes her mistake in part (b) and instead states, “The probability that the true mean is between 8.4 and 8.8 is 0.95.” What misinterpretation is she making now?
(d) Finally, in her defense for using the Normal distribution to determine the confidence interval she says, “Because the sample size is quite large, the population of alumni ratings will be approximately Normal.” Explain to Ima her misunderstanding and correct this statement.