In a 1868 paper, German physician Carl Wunderlich reported that the mean body temperature for healthy adults is 98.6° F, based on over a million body temperature readings. 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.

To find the value of the test statistic, we first need to calculate the sample mean and standard deviation. The sample mean is the average of the body temperatures, which can be calculated by summing all the temperatures and dividing by the number of observations. In this case, the sample mean is (98.3 + 98.4 + 98.5 + 98.6 + 97.3 + 98.4 + 98.1) / 7 = 98.3°F.

The next step is to calculate the sample standard deviation, which measures the spread of the body temperatures. We can use the formula:

standard deviation = sqrt((sum of (body temperature - sample mean)^2) / (number of observations - 1))

Using this formula, we can calculate the standard deviation for the given data:

(98.3 - 98.3)^2 + (98.4 - 98.3)^2 + (98.5 - 98.3)^2 + (98.6 - 98.3)^2 + (97.3 - 98.3)^2 + (98.4 - 98.3)^2 + (98.1 - 98.3)^2 / (7 - 1) = 0.0661

With the sample mean and standard deviation calculated, we can use these values to find the test statistic. In this case, we are comparing the sample mean (98.3°F) to the generally accepted mean (98.6°F) reported by Carl Wunderlich. The test statistic can be calculated as:

test statistic = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(number of observations)

Plugging in the values, we have (98.3 - 98.6) / (0.0661 / sqrt(7)) = -0.3 / (0.0661 / 2.6458) ≈ -4.54.

Therefore, the value of the test statistic is approximately -4.54.

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