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To find the p-value and the significance level, we need to perform a hypothesis test. Let's assume we are testing the hypothesis that the population mean GPA is different from a certain value (let's say 3.0 for this example).
1. **Hypothesis Setup:**
- Null Hypothesis (\(H_0\)): \(\mu = 3.0\)
- Alternative Hypothesis (\(H_a\)): \(\mu \neq 3.0\)
2. **Test Statistic:**
We use the t-test for the sample mean since the population standard deviation is unknown. The test statistic is calculated as follows:
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\]
Where:
- \(\bar{x} = 3.14\) (sample mean)
- \(\mu = 3.0\) (hypothesized population mean)
- \(s = 0.07\) (sample standard deviation)
- \(n = 40\) (sample size)
Plugging in the values:
\[
t = \frac{3.14 - 3.0}{0.07 / \sqrt{40}} = \frac{0.14}{0.01107} \approx 12.64
\]
3. **Degrees of Freedom:**
The degrees of freedom (df) for this test is \( n - 1 = 40 - 1 = 39 \).
4. **P-Value:**
Using a t-distribution table or a calculator, we find the p-value corresponding to the calculated t-value (12.64) with 39 degrees of freedom.
Given the large t-value, the p-value will be extremely small (close to 0). For practical purposes, it is often reported as \( p < 0.0001 \).
5. **Significance Level:**
The significance level (\(\alpha\)) is typically chosen by the researcher before conducting the test. Common significance levels are 0.05, 0.01, or 0.001.
If not specified, we can assume a common significance level such as 0.05.
**Conclusion:**
- The p-value is extremely small ( \( p < 0.0001 \) ).
- If the significance level is 0.05, we reject the null hypothesis since the p
1. **Hypothesis Setup:**
- Null Hypothesis (\(H_0\)): \(\mu = 3.0\)
- Alternative Hypothesis (\(H_a\)): \(\mu \neq 3.0\)
2. **Test Statistic:**
We use the t-test for the sample mean since the population standard deviation is unknown. The test statistic is calculated as follows:
\[
t = \frac{\bar{x} - \mu}{s / \sqrt{n}}
\]
Where:
- \(\bar{x} = 3.14\) (sample mean)
- \(\mu = 3.0\) (hypothesized population mean)
- \(s = 0.07\) (sample standard deviation)
- \(n = 40\) (sample size)
Plugging in the values:
\[
t = \frac{3.14 - 3.0}{0.07 / \sqrt{40}} = \frac{0.14}{0.01107} \approx 12.64
\]
3. **Degrees of Freedom:**
The degrees of freedom (df) for this test is \( n - 1 = 40 - 1 = 39 \).
4. **P-Value:**
Using a t-distribution table or a calculator, we find the p-value corresponding to the calculated t-value (12.64) with 39 degrees of freedom.
Given the large t-value, the p-value will be extremely small (close to 0). For practical purposes, it is often reported as \( p < 0.0001 \).
5. **Significance Level:**
The significance level (\(\alpha\)) is typically chosen by the researcher before conducting the test. Common significance levels are 0.05, 0.01, or 0.001.
If not specified, we can assume a common significance level such as 0.05.
**Conclusion:**
- The p-value is extremely small ( \( p < 0.0001 \) ).
- If the significance level is 0.05, we reject the null hypothesis since the p
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