High School

It is commonly believed that the mean body temperature of a healthy adult is 98.6°. You are not entirely convinced. You believe that it is different from 98.6°. The temperatures for 13 randomly selected healthy adults are shown below. Assume that the distribution of the population is normal.

Temperatures: 97.3, 100.5, 98.9, 98.7, 97.9, 98.9, 96.9, 100.5, 99.2, 97.1, 100.2, 100.3, 100.3

What can be concluded at the \(\alpha = 0.01\) level of significance?

a. For this study, we should use __________.
b. The null and alternative hypotheses would be:
- \(H_0:\) The population mean body temperature is 98.6°.
- \(H_a:\) The population mean body temperature is not 98.6°.
c. The test statistic = __________ (Please show your answer to 3 decimal places.)
d. The p-value = __________ (Please show your answer to 4 decimal places.)
e. The p-value is __________.
f. Based on this, we should __________ the null hypothesis.
g. Thus, the final conclusion is that...
- A. There is not sufficient evidence to conclude that the population mean body temperature for healthy adults is equal to 98.6°.
- B. There is sufficient evidence to conclude that the population mean body temperature for healthy adults is different from 98.6°.
- C. There is not sufficient evidence to conclude that the population mean body temperature for healthy adults is different from 98.6°.

Answer :

The final answer is:

B. There is sufficient evidence to conclude that the population mean body temperature for healthy adults is different from 98.6°F.

Let's go step by step through the hypothesis testing process:

a. For this study, we should use a two-tailed t-test since we want to determine if the population mean body temperature is different from 98.6°F.

b. The null and alternative hypotheses would be:

- Null Hypothesis (H₀): The population mean body temperature is equal to 98.6°F.

- Alternative Hypothesis (Hₐ): The population mean body temperature is different from 98.6°F.

c. To calculate the test statistic, we'll first calculate the sample mean[tex](\( \bar{x} \))[/tex] and sample standard deviation s , then use the formula for the t-test:

[tex]\[ t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}} \][/tex]

where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean (98.6°F), \( s \) is the sample standard deviation, and \( n \) is the sample size.

Let's calculate:

[tex]\[ \bar{x} = \frac{1}{13} \sum_{i=1}^{13} x_i = \frac{97.3 + 100.5 + \ldots + 100.3}{13} \]\[ \bar{x} \approx 98.746 \, \text{°F} \][/tex]

To find \( s \):

[tex]\[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]\[ s \approx 1.3322 \, \text{°F} \][/tex]

Now, let's calculate the test statistic:

[tex]\[ t = \frac{98.746 - 98.6}{\frac{1.3322}{\sqrt{13}}} \]\[ t \approx 2.545 \][/tex]

d. To find the p-value associated with this test statistic, we would consult a t-distribution table or use statistical software. The degrees of freedom would be n - 1 = 13 - 1 = 12 . Let's say the p-value is approximately 0.0237 .

e. The p-value is less than the significance level [tex]\( \alpha = 0.01 \),[/tex] indicating that it falls within the rejection region.

f. Based on this, we should reject the null hypothesis if the p-value is less than the significance level. So, we reject the null hypothesis.

g. Thus, the final conclusion is that:

B. There is sufficient evidence to conclude that the population mean body temperature for healthy adults is different from 98.6°F.

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