14.5B

Here are the IQ test scores of 31 seventh-grade girls in a Midwest school district:

$\begin{array}{rrrrrrrrrrr} 114 & 100 & 104 & 89 & 102 & 91 & 114 & 114 & 103 & 105 & \\ 108 & 130 & 120 & 132 & 111 & 128 & 118 & 119 & 86 & 72 & \\ 111 & 103 & 74 & 112 & 107 & 103 & 98 & 96 & 112 & 112 & 93 \end{array}$

These 31 girls are an SRS of all seventh-grade girls in the school district. Suppose that the standard deviation of IQ scores in this population is known to be $\sigma=15.$ . A stemplot of the distribution of these 31 scores shows that there are no major departures from Normality, and thus we assume the distribution of the scores to be close to Normal. Estimate the mean IQ score for all seventh-grade girls in the school district, using a 99% confidence interval.

Solution

The conditions for inference are satisfied by the statement of the problem. The formula for estimation of the mean score $\mu$ for all seventh-grade girls in the school with a determined confidence interval is given by:

$\bar{x} \pm z^\ast \displaystyle{ \frac{\sigma}{\sqrt{n}} },$

where $n=31$ and $\sigma=15$ are directly given in the body of the problem. The mean of the SRS is $\bar{x}=102.612903.$ To compute the critical value $z^\ast$ for confidence of 99%, we look up table C: $z^\ast = 2.576.$ The corresponding interval is then