1 TIMEREMAINING \( 49: 59 \) A researcher collated data on Americans' leisure time activities. She found the mean nümber of hours spent watching television each weekday to be 2.7 hours with a standard deviation of 0.2 hours. Jonathan believes that his football team buddies watch less television than the average American. He gathered data from 40 football teammates and found the mean to be 2.3. Which of the following are the correct null and alternate hypotheses? \( \mathrm{H}_{0}: \mu=2.7 ; \mathrm{H}_{\mathrm{a}}: \mu<2.7 \) \( \mathrm{H}_{0}: \mu \neq 2.7 ; \mathrm{H}_{\mathrm{a}}: \mu=2.3 \) \( \mathrm{H}_{0}: \mu=2.7 ; \mathrm{H}_{\mathrm{a}}: \mu \neq 2.7 \) \( \mathrm{H}_{0}: \mu<2.7 ; \mathrm{H}_{\mathrm{a}}: \mu \geq 2.3 \)

In hypothesis testing, the null hypothesis (\(H_0\)) is typically a statement of no effect or no difference, while the alternative hypothesis (\(H_a\)) represents what we are trying to prove. In this case, Jonathan's argument suggests that he believes his teammates watch less television than the average American, so the correct hypotheses would be \(H_0: \mu = 2.7\) (no difference) and \(H_a: \mu < 2.7\) (indicating his belief). With Jonathan's sample size of 40 teammates, he can use a t-test to compare his sample mean with the population mean. To avoid common pitfalls, make sure to check that his data meets the assumptions of normality and independence, particularly since he’s dealing with a relatively small sample. This will help ensure his results are valid and meaningful!