The average house has 13 paintings on its walls. Is the mean smaller for houses owned by teachers? The data show the results of a survey of 12 teachers who were asked how many paintings they have in their houses. Assume that the distribution of the population is normal. \[ 11,14,12,14,12,14,12,12,13,13,14,10 \] What can be concluded at the $\alpha=0.01$ level of significance? a. For this study, we should use t-test for a population mean $\sigma^{\circ}$ b. The null and alternative hypotheses would be: $0^{s}$ $0^{s}$ $0^{6}$ $0^{s}$ $\sigma^{\circ}$ c. The test statistic $t \quad \hat{\nabla} \vee=$ (please show your answer to 3 decimal places.) $\delta^{\infty}$ d. The $p$-value $=$ (Please show your answer to 4 decimal places.) e. The p-value is $\alpha$

Step 1 ：We are given a sample of 12 teachers' houses and their respective number of paintings. We are to perform a t-test for a population mean at a significance level of 0.01.

Step 2 ：The null hypothesis is that the mean number of paintings in teachers' houses is equal to 13, while the alternative hypothesis is that the mean number of paintings in teachers' houses is less than 13.

Step 3 ：We calculate the sample mean and standard deviation. The sample mean (\(\bar{x}\)) is approximately 12.583 and the standard deviation (s) is approximately 1.311.

Step 4 ：We use these to calculate the t-statistic. The t-statistic (t) is approximately -1.101.

Step 5 ：We then find the p-value associated with this t-statistic. The p-value is approximately 0.1473.

Step 6 ：We compare the p-value to the significance level to determine if we can reject the null hypothesis. Since the p-value is greater than the significance level of 0.01, we fail to reject the null hypothesis.

Step 7 ：This means that we do not have enough evidence to conclude that the mean number of paintings in teachers' houses is less than the average number of paintings in a house.

Step 8 ：Final Answer: \(\boxed{\text{a. For this study, we should use t-test for a population mean.}}\)

Step 9 ：\(\boxed{\text{b. The null and alternative hypotheses would be:}}\) \(H_0: \mu = 13\) and \(H_1: \mu < 13\)

Step 10 ：\(\boxed{\text{c. The test statistic t = -1.101}}\)

Step 11 ：\(\boxed{\text{d. The p-value = 0.1473}}\)

Step 12 ：\(\boxed{\text{e. The p-value is greater than }\alpha\text{, so we fail to reject the null hypothesis.}}\)

Step 13 ：Therefore, we do not have enough evidence to conclude that the mean number of paintings in teachers' houses is less than the average number of paintings in a house at the \(\alpha=0.01\) level of significance.