Question

A university claims that the mean time professors are in their offices for students is at least 6.5 hours each week. A random sample of eight professors finds that the mean time in their offices is 6.2 hours each week. With a sample standard deviation of 0.49 hours from a normally distributed data set, can the university’s claim be supported at α=0.05?

Answer 1

Answer with explanation:

Let $\mu$ be the population mean.

Null hypothesis : $H_0: \mu\geq 6.5$

Alternative hypothesis : $H_a: \mu<6.5$

Since alternative hypothesis is left-tailed , so the test is a left-tailed test.

Since n= 8 <30 , so we use t-test.

Test statistic for population mean : $t=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}$

$t=\dfrac{6.2-6.5}{\dfrac{0.49}{\sqrt{8}}}\approx-1.73$

Critical value for t=$t_{n-1, \alpha}=t_{7,0.05}=1.895$

Since the absolute t-value (1.73) is less than the critical t-value(1.895), it means we are fail to reject the null hypothesis.

Thus , we conclude that we have enough evidence to support the university’s claim.