所屬科目：研究所、轉學考(插大)◆統計學
1. Let X be a continuous random variable and $F(x) = F(X \leq x)$, where x is a real number. Which one of the following statements is true? (A) \( F(x) \) is non-decreasing in \( x \) (B) \( \sum_{x} F(x) = 1 \) (C) \( x\geq 0 \) (D) \( \sum_{x} x = 1 \) (E) \( F(0) = 0.5 \)
2. 100 students participated in a statistics test. Test scores are shown a normal distribution with mean 50 and standard deviation 10. Approximately how many students whose scores are between 40 and 60? (A)34 (B) 50 (C) 68 (D) 75 (E) 95
3. In an instant lottery, your chances of winning are 0.2. If you play the lottery five times and outcomes are independent, the probability that you win at least once is (A) 0.0819 (B) 0.6723 (C) 0.4096 (D) 0.7373 (E) 0.8125
4. The time to complete a standardized exam is approximately normal with a mean of 70 minutes and a standard deviation of 10 minutes. Using the 68-95-99.7 rule, what percent of students will complete the exam in under an hour? (A) \( 68\% \) (B) \( 32\% \) (C) \( 16\% \) (D) \( 5\% \) (E) \( 1\% \)
5. Assume that you have a binomial experiment with $p = 0.5$ and a sample size of 100. The expected value of this distribution is (A) 0.50 (B) 0.30 (C) 100 (D) 50 (E) 25
6. A standard normal distribution is a normal distribution (A) With a mean of 1 and a standard deviation of 0 (B) With a mean of 0 and a standard deviation of 1 (C) With any mean and a standard deviation of 1 (D) With any mean and any standard deviation (E) With a mean of 0 and a standard deviation of 0
7. The probability distribution of random variable, Xᵢ is defined as follows: What is the missing value? (A) 0 (B) 0.1 (C) 0.2 (D) 0.3 (E) 0.4
8. A box contains 10 bulbs, 2 of which are defective. If 3 bulbs are picked randomly, what is the probability that at least one defective is picked? (A) \( \frac{1}{6} \) (B) \( \frac{8}{15} \) (C) \( \frac{2}{3} \) (D) \( \frac{7}{15} \) (E) \( \frac{5}{6} \)
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 \( = \frac{\alpha}{2} \)
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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