15. A professor of statistics refutes the claim that the average student spends 3 hours studying for the midterm exam. The professor thinks the students spend more time than that. Which hypotheses are used to test the claim?
(A) \( H_0: \mu \leq 3 \) vs. \( H_a: \mu > 3 \)
(B) \( H_0: \mu = 3 \) vs. \( H_a: \mu \neq 3 \)
(C) \( H_0: \mu \neq 3 \) vs. \( H_a: \mu = 3 \)
(D) \( H_0: \mu = 3 \) vs. \( H_a: \mu < 3 \)
(E) \( H_0: \mu \geq 3 \) vs. \( H_a: \mu = 3 \)

16. We test the null hypothesis $H_0: \mu=10$ and the alternative $H_a: \mu<10$, for a normal population with $\sigma=4$. A random sample of 16 observations is drawn from the population and we find the sample mean of these observations is 12. The $p$-value is closest to (A) 0.0228 (B) 0.0456 (C) 0.1016 (D) 0.9772 (E) 0.9999

17. The nicotine content in cigarettes of a certain brand is normally distributed, with mean (in milligrams) and standard deviation $\sigma = 0.1$. The brand advertises that the mean nicotine content of its cigarettes is 1.5, but measurements on a random sample of 100 cigarettes of this brand give a mean of $\overline{x} = 1.53$. Is this evidence that the mean nicotine content is actually higher than advertised? To answer this, test the hypotheses $H_0: \mu = 1.5, H_a: \mu > 1.5$ at the 5% significance level. You conclude (A) That \( H_0 \) should be rejected. (B) That \( H_0 \) should not be rejected. (C) That \( H_{a} \) should be rejected. (D) There is a \( 5\% \) chance that the null hypothesis is true. (E) There is a \( 5\% \) chance that the alternative hypothesis is true.

18. A certain population follows a normal distribution with mean $\mu$ and standard deviation $\sigma = 2.5$. You collect data and test the hypotheses $H_0: \mu = 1, H_a: \mu \neq 1$. You obtain a p-value of 0.022. Which of the following is true? (A) A \( 90\% \) confidence interval for \( \mu \) will include the value 1. (B) A \( 95\% \) confidence interval for \( \mu \) will include the value 1. (C) A \( 95\% \) confidence interval for \( \mu \) will include the value 0. (D) A \( 99\% \) confidence interval for \( \mu \) will include the value 1. (E) A \( 99\% \) confidence interval for \( \mu \) will include the value 0.

19. Suppose that the significant level of a statistical hypothesis test is $\alpha$. The null hypothesis will be rejected when: (A) \( p \)-value \( \leq \alpha \) (B) \( p \)-value \( >\alpha \) (C) \( p \)-value \( \leq \alpha / 2 \) (D) \( p \)-value \( >\alpha / 2 \) (E) \( p \)-value \( = \frac{\alpha}{2} \)

20. The average monthly rent for one-bedroom apartments in Chattanooga has been $700. Because of the downturn in the real estate market, it is believed that there has been a decrease in the average rental. The correct hypotheses to be tested are (A) \( H_0: \mu \geq 700 \) v.s. \( H_a: \mu < 700 \) (B) \( H_0: \mu = 700 \) v.s. \( H_a: \mu \neq 700 \) (C) \( H_0: \mu > 700 \) v.s. \( H_a: \mu \leq 700 \) (D) \( H_0: \mu < 700 \) v.s. \( H_a: \mu \geq 700 \) (E) \( H_0: \mu < 700 \) v.s. \( H_a: \mu > 700 \) 

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