In an 1868 paper, German physician Carl Wunderlich reported, based on over a million body temperature readings, that the mean body temperature for healthy adults is 98.6°F. However, it is now commonly believed that the mean body temperature of a healthy adult is less than what was reported in that paper.

To test this hypothesis, a researcher measures the following body temperatures from a random sample of healthy adults: 98.3, 98.4, 98.5, 98.6, 97.3, 98.4, 98.1.

(a) Find the value of the test statistic.
(b) Find the 1% critical value.
(c) Work through this example on Minitab, and then use Minitab to find the p-value for the relevant hypothesis test.
(d) Since the sample size is less than 30, the above analysis requires that the population follows a normal distribution. What method could be used to check this assumption?

In this question, we are given a sample of body temperatures from healthy adults to test whether the mean body temperature is less than the reported value. We find the test statistic, the 1% critical value, and discuss how to find the p-value using Minitab. We also explore a method to check if the population follows a normal distribution.

Explanation:

(a) To find the value of the test statistic, we need to calculate the sample mean and standard deviation. The sample mean is the average of the body temperatures: (98.3 + 98.4 + 98.5 + 98.6 + 97.3 + 98.4 + 98.1) / 7 = 98.3857. The sample standard deviation is a measure of the variability in the data: sqrt(((98.3-98.3857)^2 + (98.4-98.3857)^2 + (98.5-98.3857)^2 + (98.6-98.3857)^2 + (97.3-98.3857)^2 + (98.4-98.3857)^2 + (98.1-98.3857)^2) / (7-1)) = 0.7201. The test statistic is calculated by subtracting the hypothesized population mean (98.6) from the sample mean and dividing by the sample standard deviation: (98.3857 - 98.6) / (0.7201 / sqrt(7)) = -1.9440.

(b) To find the 1% critical value, we need to use a t-distribution with (n-1) degrees of freedom. Since the sample size is 7, we have 7-1 = 6 degrees of freedom. Looking up the critical value in the t-distribution table, we find that the 1% critical value is -3.7074.

(c) To find the p-value for the relevant hypothesis test using Minitab, we would need to perform a t-test. However, since this is a text-based platform, we cannot go through the process of using Minitab. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic, assuming the null hypothesis is true. It can be calculated using statistical software or by consulting a t-distribution table.

(d) To check the assumption that the population follows a normal distribution, we can use a normality test. One common test is the Shapiro-Wilk test, which tests the null hypothesis that the data is normally distributed. If the p-value of the test is less than a chosen significance level (e.g., 0.05), we reject the null hypothesis and conclude that the data is not normally distributed.

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