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 \(\mu\).