sleeping outlier: a simple random sample of nine college freshmen were asked how many hours of sleep they…

Explanation:

Step1: Remove the outlier

The outlier is 24. The remaining data set is (8.5,6.5,8,9,6,7.5,8.5,9.5).

Step2: Calculate the sample - mean (\bar{x})

The formula for the sample - mean is (\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}). Here, (n = 8), (\sum_{i=1}^{8}x_{i}=8.5 + 6.5+8 + 9+6 + 7.5+8.5+9.5=64), so (\bar{x}=\frac{64}{8}=8).

Step3: Calculate the sample - standard deviation (s)

The formula for the sample - standard deviation is (s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}). ((8.5 - 8)^{2}=0.25), ((6.5 - 8)^{2}=2.25), ((8 - 8)^{2}=0), ((9 - 8)^{2}=1), ((6 - 8)^{2}=4), ((7.5 - 8)^{2}=0.25), ((8.5 - 8)^{2}=0.25), ((9.5 - 8)^{2}=2.25). (\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=0.25+2.25 + 0+1+4+0.25+0.25+2.25 = 10.5). (s=\sqrt{\frac{10.5}{7}}\approx1.225).

Step4: Determine the critical value (t_{\alpha/2})

For a 90% confidence interval and (n-1 = 7) degrees of freedom, (\alpha=1 - 0.90 = 0.10), (\alpha/2=0.05). From the (t) - distribution table, (t_{0.05,7}=1.895).

Step5: Calculate the margin of error (E)

The formula for the margin of error for a confidence interval of the mean (when the population standard - deviation (\sigma) is unknown) is (E = t_{\alpha/2}\frac{s}{\sqrt{n}}). (E=1.895\times\frac{1.225}{\sqrt{8}}\approx1.895\times0.433\approx0.82).

Step6: Construct the confidence interval

The confidence interval is (\bar{x}-E<\mu<\bar{x} + E). (8 - 0.82<\mu<8 + 0.82), so (7.18<\mu<8.82).

Answer:

(7.18<\mu<8.82)