Question

It has long been stated that the mean temperature of humans is 98.6 degrees Fahrenheit. However, two researchers currently involved in the subject thought that the mean temperature of humans is less than 98.6 degrees Fahrenheit. They measured the temperatures of 148 healthy adults 1 to 4 times daily for 3 days, obtaining 500 measurements. The sample data resulted in a sample mean of 98.5 degrees F and a sample standard deviation of 0.8 degrees Fahrenheit.Using the classical approach, judge, whether the mean temperature of humans is less than 98.6 degrees Fahrenheit at the alpha, equals 0.01 level of significance?

Answer 1

Using the classical approach with a sample mean of 98.5 degrees F, a sample standard deviation of 0.8 degrees F, and a sample size of 500, the t-value is calculated. If this t-value is less than the critical value from the t-distribution table at a 0.01 level of significance, we can conclude that the mean body temperature is lower than 98.6 degrees F.

Step-by-step explanation:

To assess whether the mean temperature of humans is less than 98.6 degrees Fahrenheit using the classical approach, we first establish the null hypothesis (H0): The mean temperature is 98.6 degrees F, and the alternative hypothesis (H1): The mean temperature is less than 98.6 degrees F.

Given the sample size (n=500), the sample mean (Øx = 98.5 degrees F), and the sample standard deviation (s = 0.8 degrees F), we can calculate the test statistic using the t-distribution because the population standard deviation is unknown. The test statistic (t) is calculated as follows:

(t = (Sample Mean - Hypothesized Mean) / (Sample Standard Deviation / sqrt(n)))
= (98.5 - 98.6) / (0.8 / sqrt(500))
= -0.1 / (0.8 / sqrt(500))
= -0.1 / 0.0358
= -2.79

The degrees of freedom (df) is n-1, which in this case is 499. With the calculated t-value and the degrees of freedom, we look up the critical t-value in the t-distribution table with a 0.01 level of significance (alpha). If our calculated t-value is less than the critical value for a one-tailed test, we reject the null hypothesis.

Since the classical approach relies on comparing the calculated t-value to the critical t-value for the specified alpha level, if the calculated test statistic lies in the rejection region (beyond the critical t-value), we conclude that there is significant evidence to support the hypothesis that the mean body temperature is lower than 98.6 degrees F.

In summary, using the sample data provided and assuming that the test statistic is beyond the critical value for t at a 0.01 level of significance, it would suggest that the average normal body temperature might indeed be less than 98.6 degrees F.