Answer :
To solve this problem, we need to conduct a hypothesis test comparing two means, where the null hypothesis is that the two population means are equal ([tex]\(H_0: \mu_1 = \mu_2\)[/tex]), and the alternative hypothesis is that the mean of the first population is greater than the mean of the second population ([tex]\(H_a: \mu_1 > \mu_2\)[/tex]). We are testing this hypothesis at a significance level of [tex]\(\alpha = 0.10\)[/tex].
Here is how the problem is approached step-by-step:
1. Calculate the Sample Means:
- For Sample #1: The mean ([tex]\(\bar{x}_1\)[/tex]) is approximately 53.443.
- For Sample #2: The mean ([tex]\(\bar{x}_2\)[/tex]) is approximately 51.613.
2. Calculate the Sample Standard Deviations:
- For Sample #1: The standard deviation (s1) is approximately 8.628.
- For Sample #2: The standard deviation (s2) is approximately 9.715.
3. Determine the Sample Sizes:
- The size of Sample #1 ([tex]\(n_1\)[/tex]) is 47.
- The size of Sample #2 ([tex]\(n_2\)[/tex]) is 60.
4. Calculate the Pooled Standard Deviation:
- The pooled standard deviation accounts for different sample sizes and is calculated as approximately 9.263.
5. Compute the Test Statistic:
- The test statistic is a t-value calculated by the formula:
[tex]\[
t = \frac{(\bar{x}_1 - \bar{x}_2)}{\text{pooled standard deviation} \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}
\][/tex]
- Inserting the known values, the test statistic is approximately 1.021.
6. Determine the Degrees of Freedom:
- The degrees of freedom for the test is [tex]\(n_1 + n_2 - 2 = 107\)[/tex].
7. Find the p-value:
- The p-value is computed using the test statistic and the degrees of freedom. It is approximately 0.1547.
8. Conclusion:
- With the p-value of approximately 0.1547 being greater than the significance level of 0.10, we do not reject the null hypothesis. This means that at the 10% significance level, there is not enough evidence to support the claim that the mean of population 1 is greater than the mean of population 2.
The final answers are:
- Test statistic: 1.018 (as reported in the text box)
- p-value: 0.1547
Remember, always compare the p-value to your significance level to draw conclusions about your hypothesis.
Here is how the problem is approached step-by-step:
1. Calculate the Sample Means:
- For Sample #1: The mean ([tex]\(\bar{x}_1\)[/tex]) is approximately 53.443.
- For Sample #2: The mean ([tex]\(\bar{x}_2\)[/tex]) is approximately 51.613.
2. Calculate the Sample Standard Deviations:
- For Sample #1: The standard deviation (s1) is approximately 8.628.
- For Sample #2: The standard deviation (s2) is approximately 9.715.
3. Determine the Sample Sizes:
- The size of Sample #1 ([tex]\(n_1\)[/tex]) is 47.
- The size of Sample #2 ([tex]\(n_2\)[/tex]) is 60.
4. Calculate the Pooled Standard Deviation:
- The pooled standard deviation accounts for different sample sizes and is calculated as approximately 9.263.
5. Compute the Test Statistic:
- The test statistic is a t-value calculated by the formula:
[tex]\[
t = \frac{(\bar{x}_1 - \bar{x}_2)}{\text{pooled standard deviation} \times \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}
\][/tex]
- Inserting the known values, the test statistic is approximately 1.021.
6. Determine the Degrees of Freedom:
- The degrees of freedom for the test is [tex]\(n_1 + n_2 - 2 = 107\)[/tex].
7. Find the p-value:
- The p-value is computed using the test statistic and the degrees of freedom. It is approximately 0.1547.
8. Conclusion:
- With the p-value of approximately 0.1547 being greater than the significance level of 0.10, we do not reject the null hypothesis. This means that at the 10% significance level, there is not enough evidence to support the claim that the mean of population 1 is greater than the mean of population 2.
The final answers are:
- Test statistic: 1.018 (as reported in the text box)
- p-value: 0.1547
Remember, always compare the p-value to your significance level to draw conclusions about your hypothesis.
