The work week for adults in the US that work full time is normally distributed with a mean of 47 hours. A newly hired engineer at a start-up company believes that employees at start-up companies work more on average then most working adults in the US. She asks 12 engineering friends at start-ups for the lengths in hours of their work week. Their responses are shown in the table below. Test the claim using a $1 \%$ level of significance. Give answer to at least 4 decimal places. \begin{tabular}{|c|} \hline Hours \\ \hline 45 \\ \hline 43 \\ \hline 53 \\ \hline 54 \\ \hline 48 \\ \hline 68 \\ \hline 57 \\ \hline 59 \\ \hline 48 \\ \hline 48 \\ \hline 50 \\ \hline 52 \\ \hline \end{tabular} a. What are the correct hypotheses? Based on the hypotheses, find the following: b. Test Statistic $=$ c. $p$-value $=$ d. The correct decision is to Select an answer e. The correct summary would be: Select an answer that the mean number of hours of all employees at start-up companies work more than the US mean of 47 hours.

Step 1 ：Formulate the null and alternative hypotheses. The null hypothesis is that the mean number of hours worked by employees at start-up companies is equal to the US mean of 47 hours. The alternative hypothesis is that the mean number of hours worked by employees at start-up companies is greater than the US mean of 47 hours.

Step 2 ：Calculate the test statistic and the p-value. The test statistic is calculated using the formula for a one-sample t-test, which is (sample mean - population mean) / (sample standard deviation / sqrt(sample size)). The p-value is then found using the t-distribution with degrees of freedom equal to sample size - 1.

Step 3 ：Make a decision based on the p-value. If the p-value is less than the level of significance (0.01 in this case), we reject the null hypothesis. If the p-value is greater than the level of significance, we fail to reject the null hypothesis.

Step 4 ：Final Answer: a. The correct hypotheses are: Null hypothesis: \(H_0: \mu = 47\) Alternative hypothesis: \(H_1: \mu > 47\) b. Test Statistic \(= \boxed{2.5688}\) c. p-value \(= \boxed{0.0131}\) d. The correct decision is to fail to reject the null hypothesis. e. The correct summary would be: There is not enough evidence to support the claim that the mean number of hours of all employees at start-up companies work more than the US mean of 47 hours.