Questions: To test for any significant difference in the number of hours between breakdowns for four machines, the following data were obtained. Machine 1 Machine 2 Machine 3 Machine 4 ------------ 6.7 8.8 10.9 9.9 8.0 7.5 10.0 12.8 5.6 9.5 9.5 12.0 7.7 10.3 9.9 10.7 8.8 9.2 8.8 11.3 7.6 9.9 8.5 11.7 (a) At the α=0.05 level of significance, what is the difference, if any, in the population mean times among the four machines? State the null and alternative hypotheses. H0: μ1=μ2=μ3=μ4 Ha: Not all the population means are equal. Find the value of the test statistic. (Round your answer to two decimal places.) Find the p-value. (Round your answer to three decimal places.) p-value = State your conclusion. - Do not reject H0. There is sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. - Reject H0. There is sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. - Do not reject H0. There is not sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. - Reject H0. There is not sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. (b) Use Fisher's LSD procedure to test for the equality of the means for machines 2 and 4. Use a 0.05 level of significance. Find the value of LSD. (Round your answer to two decimal places.) LSD= Find the pairwise absolute difference between sample means for machines 2 and 4. mean2-mean4 What conclusion can you draw after carrying out this test? - There is a significant difference between the means for machines 2 and 4. - There is not a significant difference between the means for machines 2 and 4.

To test for any significant difference in the number of hours between breakdowns for four machines, the following data were obtained. Machine 1 Machine 2 Machine 3 Machine 4 ------------ 6.7 8.8 10.9 9.9 8.0 7.5 10.0 12.8 5.6 9.5 9.5 12.0 7.7 10.3 9.9 10.7 8.8 9.2 8.8 11.3 7.6 9.9 8.5 11.7 (a) At the α=0.05 level of significance, what is the difference, if any, in the population mean times among the four machines? State the null and alternative hypotheses. H0: μ1=μ2=μ3=μ4 Ha: Not all the population means are equal. Find the value of the test statistic. (Round your answer to two decimal places.) Find the p-value. (Round your answer to three decimal places.) p-value = State your conclusion. - Do not reject H0. There is sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. - Reject H0. There is sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. - Do not reject H0. There is not sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. - Reject H0. There is not sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. (b) Use Fisher's LSD procedure to test for the equality of the means for machines 2 and 4. Use a 0.05 level of significance. Find the value of LSD. (Round your answer to two decimal places.) LSD= Find the pairwise absolute difference between sample means for machines 2 and 4. mean2-mean4 What conclusion can you draw after carrying out this test? - There is a significant difference between the means for machines 2 and 4. - There is not a significant difference between the means for machines 2 and 4.
Transcript text: To test for any significant difference in the number of hours between breakdowns for four machines, the following data were obtained. \begin{tabular}{|c|c|c|c|} \hline \begin{tabular}{c} Machine \\ $\mathbf{1}$ \end{tabular} & \begin{tabular}{c} Machine \\ $\mathbf{2}$ \end{tabular} & \begin{tabular}{c} Machine \\ $\mathbf{3}$ \end{tabular} & \begin{tabular}{c} Machine \\ $\mathbf{4}$ \end{tabular} \\ \hline 6.7 & 8.8 & 10.9 & 9.9 \\ \hline 8.0 & 7.5 & 10.0 & 12.8 \\ \hline 5.6 & 9.5 & 9.5 & 12.0 \\ \hline 7.7 & 10.3 & 9.9 & 10.7 \\ \hline 8.8 & 9.2 & 8.8 & 11.3 \\ \hline 7.6 & 9.9 & 8.5 & 11.7 \\ \hline \end{tabular} (a) At the $\alpha=0.05$ level of significance, what is the difference, if any, in the population mean times among the four machines? State the null and alternative hypotheses. $H_{0}: \mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}$ $H_{a}:$ Not all the population means are equal. Find the value of the test statistic. (Round your answer to two decimal places.) $\square$ Find the $p$-value. (Round your answer to three decimal places.) $p$-value $=$ $\square$ State your conclusion. Do not reject $H_{0}$. There is sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. Reject $H_{0}$. There is sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. Do not reject $H_{0}$. There is not sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. Reject $H_{0}$. There is not sufficient evidence to conclude that the mean time between breakdowns is not the same for the four machines. (b) Use Fisher's LSD procedure to test for the equality of the means for machines 2 and 4. Use a 0.05 level of significance. Find the value of LSD. (Round your answer to two decimal places.) $\mathrm{LSD}=$ $\square$ Find the pairwise absolute difference between sample means for machines 2 and 4. \[ \left|\bar{x}_{2}-\bar{x}_{4}\right|= \] $\square$ What conclusion can you draw after carrying out this test? There is a significant difference between the means for machines 2 and 4. There is not a significant difference between the means for machines 2 and 4.
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Solution

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Solution Steps

Step 1: Initial Hypothesis Testing with ANOVA

To test the null hypothesis that all group means are equal, we perform ANOVA. Calculated F-statistic: 16.176 P-value: 0 Significance level (\(lpha\)): 0.05 Since the p-value is less than \(lpha\), we reject the null hypothesis. This indicates a significant difference among the group means.

Step 2: Post-hoc Pairwise Comparisons

Using Fisher's LSD method for pairwise comparisons. Least Significant Difference (LSD): 0.659 Difference between group 1 and group 2: 1.8 The difference between group 1 and group 2 is significant. Difference between group 1 and group 3: 2.2 The difference between group 1 and group 3 is significant. Difference between group 1 and group 4: 4 The difference between group 1 and group 4 is significant. Difference between group 2 and group 3: 0.4 The difference between group 2 and group 3 is not significant. Difference between group 2 and group 4: 2.2 The difference between group 2 and group 4 is significant. Difference between group 3 and group 4: 1.8 The difference between group 3 and group 4 is significant.

Final Answer:

Based on the ANOVA and post-hoc tests, we have identified the pairs of groups with statistically significant differences in their mean values.

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