Question

the dean of a university estimates that the mean number of classroom hours per week for full - time faculty is 11.0. as a member of the student council, you want to test this claim. a random sample of the number of classroom hours for eight full - time faculty for one week is shown in the table below. at \\( \alpha=0.05 \\), can you reject the deans claim? complete parts (a) through (d) below. assume the population is normally distributed. 10.4 9.4 13.1 7.2 7.7 9.2 13.1 8.6 (a) write the claim mathematically and identify \\( h_{0} \\) and \\( h_{a} \\). which of the following correctly states \\( h_{0} \\) and \\( h_{a} \\)? a. \\( h_{0}: \mu \leq 11.0 \\) \\( h_{a}: \mu>11.0 \\) b. \\( h_{0}: \mu<11.0 \\) \\( h_{a}: \mu \geq 11.0 \\) c. \\( h_{0}: \mu>11.0 \\) \\( h_{a}: \mu \leq 11.0 \\) d. \\( h_{0}: \mu = 11.0 \\) \\( h_{a}: \mu \
eq 11.0 \\) e. \\( h_{0}: \mu \geq 11.0 \\) \\( h_{a}: \mu<11.0 \\) f. \\( h_{0}: \mu \
eq 11.0 \\) \\( h_{a}: \mu = 11.0 \\) (b) use technology to find the p - value. \\( p = \\) (round to three decimal places as needed.)

Explanation:

Step1: Calculate sample mean and sample standard deviation

Using a calculator or software (e.g., in Excel, use AVERAGE and STDEV.S functions on the data set \(10.4,9.4,13.1,7.2,7.7,9.2,13.1,8.6\)).
The sample mean \(\bar{x}=\frac{10.4 + 9.4+13.1+7.2+7.7+9.2+13.1+8.6}{8}=\frac{78.7}{8}=9.8375\)
The sample standard deviation \(s=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}\approx2.77\)

Step2: Calculate the t - statistic

The formula for the t - statistic in a one - sample t - test is \(t=\frac{\bar{x}-\mu}{s/\sqrt{n}}\), where \(\mu = 11.0\), \(n = 8\), \(\bar{x}=9.8375\), \(s\approx2.77\)
\(t=\frac{9.8375 - 11.0}{2.77/\sqrt{8}}\approx\frac{- 1.1625}{0.979}\approx - 1.19\)

Step3: Calculate the P - value

Since this is a two - tailed test (\(H_{a}:\mu
eq11.0\)), and the degrees of freedom \(df=n - 1=8 - 1 = 7\)
Using a t - distribution table or technology (e.g., in R: \(2pt(-1.19,7)\) or in Excel: \(2T.DIST(-1.19,7,TRUE)\))
The P - value is approximately \(0.279\)

Answer:

\(P = 0.279\)