Step 1 :Given in the problem, we have: \(\bar{X} = 98.4\degree F\), \(\mu = 98.6\degree F\), \(s = 1\degree F\), and \(n = 275\).
Step 2 :We will use the formula for the t-test statistic, which is: \(t = \frac{\bar{X} - \mu}{s / \sqrt{n}}\).
Step 3 :Substituting these values into the formula, we get: \(t = \frac{98.4 - 98.6}{1 / \sqrt{275}} = -0.2 / (1 / \sqrt{275})\).
Step 4 :Calculating the denominator first: \(1 / \sqrt{275} \approx 0.0603\).
Step 5 :Then substitute this into the equation: \(t = -0.2 / 0.0603 \approx -3.32\).
Step 6 :So, the test statistic \(t_0\) is approximately \(-3.32\) (rounded to two decimal places).
Step 7 :Next, we need to find the P-value. The P-value is the probability of obtaining a result as extreme as the observed data, assuming that the null hypothesis is true.
Step 8 :Since we are conducting a one-tailed test (we are testing whether the mean temperature is less than \(98.6\degree F\)), we will look up the P-value corresponding to our t statistic in a t-distribution table or use a statistical software or calculator.
Step 9 :Unfortunately, without a t-distribution table or software, we cannot calculate the exact P-value here. However, given that our t statistic is \(-3.32\) and we have a large sample size (275), we can say that our P-value will be very small and likely less than our significance level of 0.01.
Step 10 :Therefore, we would reject the null hypothesis and conclude that the mean temperature of humans is less than \(98.6\degree F\).
Step 11 :The final answer is \(\boxed{-3.32}\).