Answer :
To solve this problem, we need to test the hypothesis regarding the mean body temperature of a population based on a sample. Here is how you can approach the problem step-by-step:
### Step 1: Set the Hypotheses
We need to determine the null and alternative hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population mean [tex]\(\mu\)[/tex] is equal to 98.6°F.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The population mean [tex]\(\mu\)[/tex] is not equal to 98.6°F.
Thus, we choose:
[tex]\(H_0: \mu = 98.6\)[/tex]
[tex]\(H_a: \mu \neq 98.6\)[/tex]
This corresponds to a two-tailed test.
### Step 2: Check Conditions
Before proceeding with the t-test, we should confirm the conditions:
- The sample is random, and the observations are independent.
- The distribution of the population is approximately Normal.
Since these conditions are mentioned as fulfilled in the problem, we can proceed with the t-test.
### Step 3: Calculate the Test Statistic
1. Sample Data:
The sample consists of body temperatures:
[tex]\[ 98.5, 98.2, 99.0, 96.3, 98.7, 98.7, 97.1, 99.1, 98.3, 97.6 \][/tex]
2. Sample Size ([tex]\(n\)[/tex]):
The sample size is 10.
3. Sample Mean ([tex]\(\bar{x}\)[/tex]):
Calculate the mean of the sample data.
4. Sample Standard Deviation ([tex]\(s\)[/tex]):
Calculate the standard deviation of the sample using Bessel's correction ([tex]\(ddof=1\)[/tex]).
5. Calculate the t-statistic:
Use the formula:
[tex]\[ t = \frac{\bar{x} - 98.6}{s/\sqrt{n}} \][/tex]
The calculation results in a t-statistic of approximately [tex]\(-1.59\)[/tex].
### Conclusion
With the calculated t-statistic in hand, you can compare it to the critical t-value from the t-distribution table (based on the degree of freedom [tex]\(n-1\)[/tex] and the significance level 0.05) to conclude whether to reject or fail to reject the null hypothesis. Since the problem only asks us to find the test statistic, we stop here. The correct choice of hypotheses based on the given options is:
F. [tex]\(H_0: \mu = 98.6\)[/tex], [tex]\(H_a: \mu \neq 98.6\)[/tex].
### Step 1: Set the Hypotheses
We need to determine the null and alternative hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): The population mean [tex]\(\mu\)[/tex] is equal to 98.6°F.
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): The population mean [tex]\(\mu\)[/tex] is not equal to 98.6°F.
Thus, we choose:
[tex]\(H_0: \mu = 98.6\)[/tex]
[tex]\(H_a: \mu \neq 98.6\)[/tex]
This corresponds to a two-tailed test.
### Step 2: Check Conditions
Before proceeding with the t-test, we should confirm the conditions:
- The sample is random, and the observations are independent.
- The distribution of the population is approximately Normal.
Since these conditions are mentioned as fulfilled in the problem, we can proceed with the t-test.
### Step 3: Calculate the Test Statistic
1. Sample Data:
The sample consists of body temperatures:
[tex]\[ 98.5, 98.2, 99.0, 96.3, 98.7, 98.7, 97.1, 99.1, 98.3, 97.6 \][/tex]
2. Sample Size ([tex]\(n\)[/tex]):
The sample size is 10.
3. Sample Mean ([tex]\(\bar{x}\)[/tex]):
Calculate the mean of the sample data.
4. Sample Standard Deviation ([tex]\(s\)[/tex]):
Calculate the standard deviation of the sample using Bessel's correction ([tex]\(ddof=1\)[/tex]).
5. Calculate the t-statistic:
Use the formula:
[tex]\[ t = \frac{\bar{x} - 98.6}{s/\sqrt{n}} \][/tex]
The calculation results in a t-statistic of approximately [tex]\(-1.59\)[/tex].
### Conclusion
With the calculated t-statistic in hand, you can compare it to the critical t-value from the t-distribution table (based on the degree of freedom [tex]\(n-1\)[/tex] and the significance level 0.05) to conclude whether to reject or fail to reject the null hypothesis. Since the problem only asks us to find the test statistic, we stop here. The correct choice of hypotheses based on the given options is:
F. [tex]\(H_0: \mu = 98.6\)[/tex], [tex]\(H_a: \mu \neq 98.6\)[/tex].