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Answer :
To solve this question, we need to perform a hypothesis test for a single sample mean. The claim we want to test is whether the population mean [tex]\(\mu\)[/tex] is greater than 87.4. We have a sample and need to calculate the test statistic and p-value based on this data.
Here’s a step-by-step solution:
1. Set Up the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 87.4\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu > 87.4\)[/tex] (This is a right-tailed test)
2. Sample Data:
- We have a set of 50 data points from the sample.
3. Calculate the Sample Mean and Standard Deviation:
- Compute the mean ([tex]\(\bar{x}\)[/tex]) of the sample data.
- Compute the sample standard deviation ([tex]\(s\)[/tex]).
4. Determine the Test Statistic:
The test statistic for a t-test when the population standard deviation is unknown is calculated as:
[tex]\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\][/tex]
where:
- [tex]\(\bar{x}\)[/tex] is the sample mean.
- [tex]\(\mu_0\)[/tex] is the mean under the null hypothesis (87.4).
- [tex]\(s\)[/tex] is the sample standard deviation.
- [tex]\(n\)[/tex] is the sample size.
5. Calculate the P-Value:
Since this is a right-tailed test, the p-value is the probability that the test statistic is greater than the calculated value.
6. Decision Making:
Compare the p-value to the significance level [tex]\(\alpha = 0.001\)[/tex]:
- If the p-value is less than or equal to [tex]\(\alpha\)[/tex], we reject the null hypothesis.
- If the p-value is greater than [tex]\(\alpha\)[/tex], we do not reject the null hypothesis.
Based on the calculations, the results are:
- Test Statistic: 0.614 (rounded to three decimal places)
- P-Value: 0.2712 (rounded to four decimal places)
- Conclusion: The p-value is greater than [tex]\(\alpha = 0.001\)[/tex], so we fail to reject the null hypothesis. Therefore, there is not enough evidence to support the claim that the population mean is greater than 87.4.
Here’s a step-by-step solution:
1. Set Up the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu = 87.4\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu > 87.4\)[/tex] (This is a right-tailed test)
2. Sample Data:
- We have a set of 50 data points from the sample.
3. Calculate the Sample Mean and Standard Deviation:
- Compute the mean ([tex]\(\bar{x}\)[/tex]) of the sample data.
- Compute the sample standard deviation ([tex]\(s\)[/tex]).
4. Determine the Test Statistic:
The test statistic for a t-test when the population standard deviation is unknown is calculated as:
[tex]\[
t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
\][/tex]
where:
- [tex]\(\bar{x}\)[/tex] is the sample mean.
- [tex]\(\mu_0\)[/tex] is the mean under the null hypothesis (87.4).
- [tex]\(s\)[/tex] is the sample standard deviation.
- [tex]\(n\)[/tex] is the sample size.
5. Calculate the P-Value:
Since this is a right-tailed test, the p-value is the probability that the test statistic is greater than the calculated value.
6. Decision Making:
Compare the p-value to the significance level [tex]\(\alpha = 0.001\)[/tex]:
- If the p-value is less than or equal to [tex]\(\alpha\)[/tex], we reject the null hypothesis.
- If the p-value is greater than [tex]\(\alpha\)[/tex], we do not reject the null hypothesis.
Based on the calculations, the results are:
- Test Statistic: 0.614 (rounded to three decimal places)
- P-Value: 0.2712 (rounded to four decimal places)
- Conclusion: The p-value is greater than [tex]\(\alpha = 0.001\)[/tex], so we fail to reject the null hypothesis. Therefore, there is not enough evidence to support the claim that the population mean is greater than 87.4.
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