A random sample of 10 independent healthy people showed the body temperatures given below​ (in degrees​ Fahrenheit). Test the hypothesis that the population mean is not 98.6 degrees°​F, using a significance level of 0.05.
98.5
98.9
99.0
96.2
98.6
98.3
97.4
99.4
98.6
97.7
 
  test statistic.

To test the hypothesis that the population mean body temperature is different from 98.6 degrees Fahrenheit, we need to perform a one-sample t-test. Here are the steps involved:

### Step 1: State the Hypotheses

- **Null Hypothesis (\(H_0\))**: The population mean body temperature is 98.6°F (\(\mu = 98.6\)).
- **Alternative Hypothesis (\(H_a\))**: The population mean body temperature is not 98.6°F (\(\mu \neq 98.6\)).

### Step 2: Collect the Data

The sample data provided is:
\[ 
98.5, 98.9, 99.0, 96.2, 98.6, 98.3, 97.4, 99.4, 98.6, 97.7 
\]

### Step 3: Calculate the Sample Mean and Standard Deviation

First, calculate the sample mean (\(\bar{x}\)) and sample standard deviation (\(s\)).

### Step 4: Calculate the Test Statistic

The formula for the test statistic in a one-sample t-test is:

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

where:
- \(\bar{x}\) is the sample mean
- \(\mu\) is the population mean under the null hypothesis (98.6°F)
- \(s\) is the sample standard deviation
- \(n\) is the sample size

Let's perform the calculations: 

1. **Sample mean \(\bar{x}\):**

\[
\bar{x} = \frac{\sum x_i}{n} = \frac{98.5 + 98.9 + 99.0 + 96.2 + 98.6 + 98.3 + 97.4 + 99.4 + 98.6 + 97.7}{10}
\]

2. **Sample standard deviation \(s\):**

\[
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
\]

3. **Sample size \(n\):** 10

4. **Test statistic \(t\):**

\[
t = \frac{\bar{x} - 98.6}{\frac{s}{\sqrt{10}}}
\]

Let's compute these values.### Calculation Results:

1. **Sample Mean (\(\bar{x}\))**: \(98.26\)
2. **Sample Standard Deviation (\(s\))**: \(0.9336\)
3. **Test Statistic (\(t\))**: \(-1.152\)

### Interpretation

The test statistic \(t = -1.152\) indicates how many standard deviations the sample mean is from the hypothesized population mean of 98.6°F. To determine whether this is statistically significant, we would compare the test statistic to a critical value from the t-distribution, or use it to calculate a p-value.

### Next Steps

1. **Determine the Degrees of Freedom**: \(df = n - 1 = 10 - 1 = 9\)
2. **Find the Critical t-value**: For a two-tailed test at the significance level \(\alpha = 0.05\) with \(df = 9\).
3. **Compare the Test Statistic**: 
   - If \(|t|\) is greater than the critical t-value, we reject the null hypothesis.
   - Otherwise, we fail to reject the null hypothesis.

Let's calculate the critical t-value and make a decision.### Decision

1. **Critical t-value**: \( \pm 2.262 \) (for \( df = 9 \) and \(\alpha = 0.05\))
2. **p-value**: \(0.279\)

### Conclusion

- **Comparison**: \(|t| = 1.152\) is less than the critical t-value \(2.262\).
- **p-value**: \(0.279\) is greater than the significance level \(0.05\).

Since the absolute value of the test statistic is less than the critical t-value and the p-value is greater than 0.05, we fail to reject the null hypothesis. There is not enough evidence to conclude that the population mean body temperature is different from 98.6°F at the 5% significance level.

---

Would you like any additional details or have any questions about this process? Here are some related questions you might find interesting:

1. What is the significance level in hypothesis testing?
2. How do you interpret a p-value in a t-test?
3. What assumptions are made when performing a t-test?
4. How does the sample size affect the test statistic?
5. What is the difference between a one-tailed and a two-tailed test?
6. Why do we use a t-distribution instead of a normal distribution in this test?
7. How would the results change if the significance level were 0.01?
8. What other tests could be used for this type of hypothesis?

**Tip**: Always check if the assumptions for the t-test (such as normality and independence of data points) are met before performing the test.

Explore how to test the hypothesis that the population mean body temperature differs from 98.6°F using a one-sample t-test. Includes steps to calculate the test statistic, interpretation of results, and decision-making process with a significance level of 0.05.

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