Questions: The following three independent random samples are obtained from three normally distributed populations with equal variances. The dependent variable is starting hourly wage, and the groups are the types of position (internship, co-op, work study). We are testing the claim that position is a factor in starting salaries for new college graduates, at α=0.1 given the following data Group 1: Internship Group 2: Co-op Group 3: Work Study 10.5 11.25 13.25 12.25 9.25 9.75 15.25 11.25 10 16.5 12.25 9.5 14.25 13 10 14.5 13.25 13.5 12.25 13.75 8.5 13.75 16.5 11.5 11.5 10 15.25 1. Your friend Gail helped you with the null and alternative hypotheses... H0: μ1=μ2=μ3 H1 : At least two of the means differ from each other. 2. For this study, we should use Select an answer O 3. The test-statistic for this data = (Please show your answer to 3 decimal places.) 4. The p-value for this sample = (Please show your answer to 4 decimal places.) 5. The p-value is select an answer o α 6. Based on this, we should (Select an answer o) hypothesis 7. As such, the final conclusion is that... There is sufficient evidence to support the claim that position is a factor in starting salaries for new college graduates. There is insufficient evidence to support the claim that position is a factor in starting salaries for new college graduates.

The following three independent random samples are obtained from three normally distributed populations with equal variances. The dependent variable is starting hourly wage, and the groups are the types of position (internship, co-op, work study). We are testing the claim that position is a factor in starting salaries for new college graduates, at α=0.1 given the following data Group 1: Internship Group 2: Co-op Group 3: Work Study 10.5 11.25 13.25 12.25 9.25 9.75 15.25 11.25 10 16.5 12.25 9.5 14.25 13 10 14.5 13.25 13.5 12.25 13.75 8.5 13.75 16.5 11.5 11.5 10 15.25 1. Your friend Gail helped you with the null and alternative hypotheses... H0: μ1=μ2=μ3 H1 : At least two of the means differ from each other. 2. For this study, we should use Select an answer O 3. The test-statistic for this data = (Please show your answer to 3 decimal places.) 4. The p-value for this sample = (Please show your answer to 4 decimal places.) 5. The p-value is select an answer o α 6. Based on this, we should (Select an answer o) hypothesis 7. As such, the final conclusion is that... There is sufficient evidence to support the claim that position is a factor in starting salaries for new college graduates. There is insufficient evidence to support the claim that position is a factor in starting salaries for new college graduates.
Transcript text: The following three independent random samples are obtained from three normally distributed populations with equal variances. The dependent variable is starting hourly wage, and the groups are the types of position (internship, co-op, work study). We are testing the claim that position is a factor in starting salaries for new college graduates, at $\alpha=0.1$ given the following data \begin{tabular}{|c|c|c|} \hline Group 1: Internship & Group 2: Co-op & Group 3: Work Study \\ \hline 10.5 & 11.25 & 13.25 \\ \hline 12.25 & 9.25 & 9.75 \\ \hline 15.25 & 11.25 & 10 \\ \hline 16.5 & 12.25 & 9.5 \\ \hline 14.25 & 13 & 10 \\ \hline 14.5 & 13.25 & 13.5 \\ \hline 12.25 & 13.75 & 8.5 \\ \hline 13.75 & 16.5 & 11.5 \\ \hline 11.5 & 10 & 15.25 \\ \hline \end{tabular} 1. Your friend Gail helped you with the null and alternative hypotheses... $H_{0}: \mu_{1}=\mu_{2}=\mu_{3}$ $H_{1}$ : At least two of the means differ from each other. 2. For this study, we should use Select an answer $O$ 3. The test-statistic for this data $=$ $\square$ (Please show your answer to 3 decimal places.) 4. The p-value for this sample $=$ $\square$ (Please show your answer to 4 decimal places.) 5. The p-value is select an answer $\qquad$ o $\alpha$ 6. Based on this, we should (Select an answer o) hypothesis 7. As such, the final conclusion is that... There is sufficient evidence to support the claim that position is a factor in starting salaries for new college graduates. There is insufficient evidence to support the claim that position is a factor in starting salaries for new college graduates.
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Solution

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Solution Steps

Step 1: Calculate \(SS_{between}\)

The sum of squares between groups is calculated as follows: \[ SS_{between} = \sum_{i=1}^k n_i (\bar{X}_i - \bar{X})^2 = 21.1435 \]

Step 2: Calculate \(SS_{within}\)

The sum of squares within groups is calculated as: \[ SS_{within} = \sum_{i=1}^k \sum_{j=1}^{n_i} (X_{ij} - \bar{X}_i)^2 = 108.8056 \]

Step 3: Calculate \(MS_{between}\)

The mean square between groups is calculated using the formula: \[ MS_{between} = \frac{SS_{between}}{df_{between}} = \frac{21.1435}{2} = 10.5718 \]

Step 4: Calculate \(MS_{within}\)

The mean square within groups is calculated as: \[ MS_{within} = \frac{SS_{within}}{df_{within}} = \frac{108.8056}{24} = 4.5336 \]

Step 5: Calculate the F-statistic

The F-statistic is calculated using the mean squares: \[ F = \frac{MS_{between}}{MS_{within}} = \frac{10.5718}{4.5336} = 2.3319 \]

Step 6: Calculate the P-value

The P-value is determined using the F-distribution: \[ P = 1 - F(F_{observed}; df_{between}, df_{within}) = 1 - F(2.3319; 2, 24) = 0.1187 \]

Step 7: Conclusion

Based on the calculated F-statistic and P-value, we can compare the P-value to the significance level \(\alpha = 0.1\) to make a decision regarding the null hypothesis.

Final Answer

  1. \(H_{0}: \mu_{1}=\mu_{2}=\mu_{3}\)
  2. ANOVA
  3. \(F = \boxed{2.332}\)
  4. \(p\text{-value} = \boxed{0.1187}\)
  5. \(p\text{-value} > \alpha\)
  6. Fail to reject the null hypothesis
  7. There is insufficient evidence to support the claim that position is a factor in starting salaries for new college graduates.
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