Answer :
To test the claim [tex]\(H_a: \mu_1 < \mu_2\)[/tex] at a significance level of [tex]\(\alpha = 0.05\)[/tex], using two samples of data, follow these steps:
1. State the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu_1 = \mu_2\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu_1 < \mu_2\)[/tex]
2. Collect Sample Data:
- Sample 1: Composed of 52 data points.
- Sample 2: Composed of 46 data points.
3. Calculate Sample Means and Standard Deviations:
- For Sample 1, compute the mean and standard deviation.
- For Sample 2, compute the mean and standard deviation.
4. Compute the Standard Error of Difference (SED):
[tex]\[
\text{SED} = \sqrt{\left(\frac{\text{std1}^2}{n1}\right) + \left(\frac{\text{std2}^2}{n2}\right)}
\][/tex]
where [tex]\(\text{std1}\)[/tex] and [tex]\(\text{std2}\)[/tex] are the standard deviations of Sample 1 and Sample 2, respectively, and [tex]\(n1\)[/tex] and [tex]\(n2\)[/tex] are their sizes.
5. Calculate the Test Statistic:
[tex]\[
t = \frac{\text{mean1} - \text{mean2}}{\text{SED}}
\][/tex]
6. Determine Degrees of Freedom (df):
- The degrees of freedom are determined to be equal to the smaller of [tex]\(n1 - 1\)[/tex] and [tex]\(n2 - 1\)[/tex].
7. Find the p-value:
- Using the calculated [tex]\(t\)[/tex]-statistic and degrees of freedom, find the [tex]\(p\)[/tex]-value from the [tex]\(t\)[/tex]-distribution for a one-tailed test.
Here is the result for this specific test:
- Test Statistic: [tex]\(-2.437\)[/tex]
- p-value: [tex]\(0.0094\)[/tex]
Conclusion:
Since the p-value ([tex]\(0.0094\)[/tex]) is less than the significance level ([tex]\(\alpha = 0.05\)[/tex]), we reject the null hypothesis [tex]\(H_0\)[/tex]. There is significant evidence to support the claim that [tex]\(\mu_1 < \mu_2\)[/tex].
1. State the Hypotheses:
- Null Hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu_1 = \mu_2\)[/tex]
- Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu_1 < \mu_2\)[/tex]
2. Collect Sample Data:
- Sample 1: Composed of 52 data points.
- Sample 2: Composed of 46 data points.
3. Calculate Sample Means and Standard Deviations:
- For Sample 1, compute the mean and standard deviation.
- For Sample 2, compute the mean and standard deviation.
4. Compute the Standard Error of Difference (SED):
[tex]\[
\text{SED} = \sqrt{\left(\frac{\text{std1}^2}{n1}\right) + \left(\frac{\text{std2}^2}{n2}\right)}
\][/tex]
where [tex]\(\text{std1}\)[/tex] and [tex]\(\text{std2}\)[/tex] are the standard deviations of Sample 1 and Sample 2, respectively, and [tex]\(n1\)[/tex] and [tex]\(n2\)[/tex] are their sizes.
5. Calculate the Test Statistic:
[tex]\[
t = \frac{\text{mean1} - \text{mean2}}{\text{SED}}
\][/tex]
6. Determine Degrees of Freedom (df):
- The degrees of freedom are determined to be equal to the smaller of [tex]\(n1 - 1\)[/tex] and [tex]\(n2 - 1\)[/tex].
7. Find the p-value:
- Using the calculated [tex]\(t\)[/tex]-statistic and degrees of freedom, find the [tex]\(p\)[/tex]-value from the [tex]\(t\)[/tex]-distribution for a one-tailed test.
Here is the result for this specific test:
- Test Statistic: [tex]\(-2.437\)[/tex]
- p-value: [tex]\(0.0094\)[/tex]
Conclusion:
Since the p-value ([tex]\(0.0094\)[/tex]) is less than the significance level ([tex]\(\alpha = 0.05\)[/tex]), we reject the null hypothesis [tex]\(H_0\)[/tex]. There is significant evidence to support the claim that [tex]\(\mu_1 < \mu_2\)[/tex].