A researcher wants to know if there is a difference between the mean amount of sleep that people get for various types of employment status. The table below shows data that was collected from a survey.
\begin{tabular}{|c|c|c|}
\hline Part Time Worker & Unemployed & Full Time Worker \\
\hline 9 & 10 & 6 \\
\hline 6 & 8 & 7 \\
\hline 9 & 8 & 6 \\
\hline 9 & 7 & 6 \\
\hline 6 & 10 & 7 \\
\hline 9 & 9 & 8 \\
\hline 7 & 9 & 6 \\
\hline 9 & 8 & 8 \\
\hline 6 & & \\
\hline
\end{tabular}

Assume that all distributions are normal, the three population standard deviations are all the same, and the data was collected independently and randomly. Use a level of significance of $\alpha=0.01$.
\[
H_{0}: \mu_{1}=\mu_{2}=\mu_{3}
\]
$H_{1}$ : At least two of the means differ from each other.
1. For this study, we should use Select an answer $v$
2. The test-statistic for this data $=$ (Please show your answer to 3 decimal places.)
3. The $p$-value for this sample $=$ (Please show your answer to 4 decimal places.)
4. The p-value is Select an answer $\alpha$
5. Base on this, we should Select an answer
6. As such, the final conclusion is that... hypothesis
There is insufficient evidence to support the claim that employment status is a factor in the amount of sleep people get.
There is sufficient evidence to support the claim that employment status is a factor in the amount of sleep people get.

Step 1 ：Calculate the means of each group: For Part Time Workers: \(\frac{9+6+9+9+6+9+7+9+6}{9} = 7.778\), For Unemployed: \(\frac{10+8+8+7+10+9+9+8}{8} = 8.625\), For Full Time Workers: \(\frac{6+7+6+6+7+8+6+8}{8} = 6.75\)

Step 2 ：Calculate the total mean: \(\frac{7.778*9 + 8.625*8 + 6.75*8}{9+8+8} = 7.725\)

Step 3 ：Calculate the sum of squares between groups (SSB): \(9*(7.778 - 7.725)^2 + 8*(8.625 - 7.725)^2 + 8*(6.75 - 7.725)^2 = 3.717\)

Step 4 ：Calculate the sum of squares within groups (SSW): \(\Sigma (value - 7.778)^2\) for Part Time Workers + \(\Sigma (value - 8.625)^2\) for Unemployed + \(\Sigma (value - 6.75)^2\) for Full Time Workers = 18.889

Step 5 ：Calculate the total sum of squares (SST): SSB + SSW = 3.717 + 18.889 = 22.606

Step 6 ：Calculate the degrees of freedom for between groups (dfB): number of groups - 1 = 3 - 1 = 2

Step 7 ：Calculate the degrees of freedom within groups (dfW): total number of values - number of groups = 25 - 3 = 22

Step 8 ：Calculate the total degrees of freedom (dfT): dfB + dfW = 2 + 22 = 24

Step 9 ：Calculate the mean squares between groups (MSB): SSB / dfB = 3.717 / 2 = 1.859

Step 10 ：Calculate the mean squares within groups (MSW): SSW / dfW = 18.889 / 22 = 0.859

Step 11 ：Calculate the test statistic (F): MSB / MSW = 1.859 / 0.859 = 2.165

Step 12 ：Find the p-value associated with this F statistic and dfB and dfW. Using a F-distribution table or a statistical software, we find: The p-value for this sample = 0.1356

Step 13 ：The p-value is greater than \(\alpha\) (0.01)

Step 14 ：Do not reject the null hypothesis because the p-value is greater than \(\alpha\)

Step 15 ：\(\boxed{\text{There is insufficient evidence to support the claim that employment status is a factor in the amount of sleep people get.}}\)