Problem
It has long been stated that the mean temperature of humans is $98.6^{\circ} \mathrm{F}$. However, two researchers currently involved in the subject thought that the mean temperature of humans is less than $98.6^{\circ} \mathrm{F}$. They measured the temperatures of 61 healthy adults 1 to 4 times dally for 3 days, obtaining 275 measurements. The samplo data resulted in a sample mean of $98.4^{\circ} \mathrm{F}$ and a sample standard deviation of $1^{\circ} \mathrm{F}$. Use the P-value approach to conduct a hypothesis test to judge whether the mean temperature of humans is less than $98.6^{\circ} \mathrm{F}$ at the $\alpha=0.01$ level of significance. State the hypotheses. \[ \begin{array}{l} H_{0}: H_{\mu}=98.6^{\circ} \mathrm{F} \\ H_{1}: \mu<98.6^{\circ} \mathrm{F} \end{array} \] Find the test statistic. \[ t_{0}=\square \] (Round to two decimal places as needed.)
Solution
Step 1 ：State the hypotheses: \(H_{0}: \mu = 98.6^{\circ}F\), \(H_{1}: \mu < 98.6^{\circ}F\)
Step 2 ：Find the test statistic using the formula: \(t_{0} = \frac{\bar{x} - \mu_{0}}{s / \sqrt{n}}\)
Step 3 ：Substitute the given values into the formula: \(t_{0} = \frac{98.4 - 98.6}{1 / \sqrt{275}}\)
Step 4 ：Calculate the test statistic: \(t_{0} = -3.32\)
Step 5 ：Final Answer: The test statistic is \(\boxed{-3.32}\)
