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.
$$\text{Unknown environment 'tabular'}$$

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: $\frac{7.778\ast 9+8.625\ast 8+6.75\ast 8}{9+8+8}=7.725$

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

Step 4 ：Calculate the sum of squares within groups (SSW): $\mathrm{\Sigma }(value-7.778{)}^2$ for Part Time Workers + $\mathrm{\Sigma }(value-8.625{)}^2$ for Unemployed + $\mathrm{\Sigma }(value-6.75{)}^2$ 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 ：$\boxed{{\displaystyle \text{There is insufficient evidence to support the claim that employment status is a factor in the amount of sleep people get.}}}$