.2 Homework
Question 10, 7.2.11
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A data set includes 109 body temperatures of healthy adult humans having a mean of $98.2^{\circ} \mathrm{F}$ and a standard deviation of $0.62^{\circ} \mathrm{F}$. Construct a $99 \%$ confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of $98.6^{\circ} \mathrm{F}$ as the mean body temperature?
Click here to view a t distribution table.
Click here to view page 1 of the standard normal distribution table.
Click here to view page 2 of the standard normal distribution table.
What is the confidence interval estimate of the population mean $\mu$ ?
$\square^{\circ} \mathrm{F}< \mu< \square^{\circ} \mathrm{F}$
(Round to three decimal places as needed.)

Step 1 ：We are given a data set of 109 body temperatures of healthy adult humans with a mean of \(98.2^\circ F\) and a standard deviation of \(0.62^\circ F\). We are asked to construct a 99% confidence interval estimate of the mean body temperature of all healthy humans.

Step 2 ：The formula for a confidence interval is \(\bar{x} \pm t_{\frac{\alpha}{2}, n-1} \cdot \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(t_{\frac{\alpha}{2}, n-1}\) is the t-score for a 99% confidence interval with 108 degrees of freedom, \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step 3 ：Substituting the given values into the formula, we get \(98.2 \pm 2.626 \cdot \frac{0.62}{\sqrt{109}}\).

Step 4 ：Calculating the margin of error, we get approximately 0.156.

Step 5 ：Subtracting and adding the margin of error from the sample mean, we get the lower and upper bounds of the confidence interval, which are approximately \(98.044^\circ F\) and \(98.356^\circ F\) respectively.

Step 6 ：This suggests that we can be 99% confident that the true mean body temperature of all healthy humans is between these two values.

Step 7 ：\(\boxed{98.044^\circ F < \mu < 98.356^\circ F}\) is the 99% confidence interval estimate of the population mean \(\mu\).