Suppose that 400 randomly selected alumni of the OSU were asked to rate the univ
ID: 3131684 • Letter: S
Question
Suppose that 400 randomly selected alumni of the OSU were asked to rate the university's counseling services on a 1 to 10 scale. The sample mean (x) was found to be 8.6. Assume that the population standard deviation is known to be sigma = 2.0. Alice computes the 95% confidence interval for the mean satisfaction score as What is her mistake? After correcting her mistake in part (a), she states "I am 95% confident that the sample mean falls between 8.404 and 8.796." What is wrong with this statement? After quickly realizing her mistake in part (b), she states "95% of alumni ratings would fall between 8.404 and 8.796." What is wrong with this statement? She realizes her mistake in part (c) and instead states "The probability that the true mean is between 8.404 and 8.796 is 0.95." What misinterpretation is she making now? Finally, in her defense for using the Normal distribution to determine the confidence coefficient, she says "Because the sample size is quite large, the population of alumni ratings will be approximately normal." Explain to Alice her misunderstanding and correct this statement.Explanation / Answer
a)
She forgot to divide the standard deviation by sqrt(n), where n = 400 here.
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b)
It is the POPULATION MEAN instead of the sample mean that is of 95% confidence inside that interval. The sample mean is definitely inside (actually, it's the center) of that interval.
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c)
It is not the alumni ratings, it is the POPULATION MEAN that is inside the confidence interval 95% of confidence intervals being arrived at using this method.
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d)
She misinterprets the confidence level as the probability of the true mean being inside the confidence interval. In fact, 95% confidence means that 95% of confidence intervals using this method on the average captures the true mean.
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e)
It is not the "population of alumni ratings", but rather, the distribution of sample means. This is by virtue of the central limit theorem.
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