9. Independently toss a fair dice 4 times. Suppose that Xᵢ is the outcome of the iᵘ toss, and X̄ = (X₁ + X₂ + X₃ + X₄) / 4. Then Var(X̄) is:
(A) \( \frac{35}{12} \)
(B) \( \frac{35}{24} \)
(C) \( \frac{35}{36} \)
(D) \( \frac{35}{48} \)
(E) \( \frac{35}{60} \)

10. As the sample size increases, the (A) Standard deviation of the population decreases (B) Population mean increases (C) Sample mean increases (D) Standard error of the sample mean increases (E) Standard error of the sample mean decreases

11. A random sample of size 25 is to be taken from a population that is normally distributed with mean 60 and standard deviation 10. The average $\overline{x}$ of the observations in our sample is to be computed. The sampling distribution of $\overline{x}$ is (A) Normal with mean 60 and standard deviation 10 (B) Normal with mean 60 and standard deviation 2 (C) Normal with mean 60 and standard deviation 0.4 (D) Normal with mean 60 and standard deviation 25 (E) Normal with mean 60 and standard deviation 2.5

12. The SAT scores of entering freshmen at a certain university have mean 1215 and standard deviation 110. A random sample of 100 freshmen is taken and $\overline{x}$, the sample mean of their SAT scores, is computed. The probability that $\overline{x}$ less than 1190 is (A) 0.0116 (B) 0.1335 (C) 0.4090 (D) 0.4562 (E) 0.1628

13. As the sample size becomes larger, the sampling distribution of the sample mean approaches a (A) Binomial distribution (B) Poisson distribution (C) Normal distribution (D) Chi-square distribution (E) t distribution

14. Which of the following is true? (A) If we draw a simple random sample of any size from any population the sampling distribution of the sample mean will be exactly Normal. (B) If we draw a simple random sample of any size from any population the sampling distribution of the sample mean will be close to Normal. (C) Central limit theorem only applies when sampling from Normal populations. (D) None of the above.

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 \( = \frac{\alpha}{2} \)

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