The mean number of sick days an employee takes per year is believed to be about 10. Members of a personnel department do not believe this figure. They randomly survey 8 employees. The number of sick days they took for the past year are as follows: 12; 5; 13; 5; 10; 8; 6; 9. Let X = the number of sick days they took for the past year. Should the personnel team believe that the mean number is about 10? Conduct a hypothesis test at the 5% level. Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.) Part (a) State the null hypothesis. O Hot A 10 OH, A # 10 O Hai # = 10 Part (c) In words, state what your random variable X represents. O X represents the average number of sick days employees take each year. O X represents the average number of employees that call out sick on a given day. O X represents the average number of employees that cal out sick for 10 days in one year. O X represents the number of sick days an employee takes in one year. ( Part (d) State the distribution to use for the test. (Enter your answer in the form z of ( where off is the degrees of freedom.) Part () What is the test statistic? (if using the z distribution round your answers to two decimal places, and if using the / distribution round your answers to three decimal places.) ..-Select-- v=Part () What is the p-value? Opvalue 0.100 Explain what the p-value means for this problem. If He is true, then there is a chance equal to the p-value that the average number of sick days for employees is at least as different from 10 as the mean of the sample is different from 10. O if He is true, then there is a chance equal to the p-value the average number of sick days for employees is not at least as different from 10 as the mean of the sample is different from 10. O If He is false, then there is a chance equal to the p-value the average number of sick days for employees is not at least as different from 10 as the mean of the sample is different from 10. O if He is false, then there is a chance equal to the p-value that the average number of sick days for employees is at least as different from 10 as the mean of the sample is different from 10. Part Col Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. 1/2(p-value) 1/2 p-value 1/2(p-value) 1/2(p-value X O O P-value p-valueB Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis], the reason for it, and write an appropriate conclusion. (0 Alpha (Enter an exact number as an integer, fraction, or decimal.) () Decision: O reject the null hypothesis O do not reject the null hypothesis (1) Reason for decision: O Since a >> p-value, we reject the null hypothesis. O Since a p-value, we do not reject the null hypothesis. (w) Conclusion: O There is sufficient evidence to conclude that the average number of sick days used per year by an employee is not equal to 10 days. O There is not sufficient evidence to conclude that the average number of sick days used per year by an employee is not equal to 10 days. Part (1) Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your answers to three decimal places.) 95% C.I