110年 - [非官方正解]110 國立臺灣大學_碩士班招生考試_國家發展研究所甲、乙組：統計學（B）#100793

科目：統計學 | 年份：110年 | 選擇題數：20 | 申論題數：4

試卷資訊

所屬科目：統計學

選擇題 (20)

1. The scores of individual students on the American College Testing (ACT) Program composite college entrance examination have a normal distribution with mean 18.6 and standard deviation 6.0. At Northside High, 36 seniors take the test. If the scores at this school have the same distribution as national scores, what is the mean of the sampling distribution of the average (sample mean) score for the 36 students? (A)1.0 (B) 3.1 (C)6.0 (D)18.6

2. I take a random sample survey of size n from a population that has mean 80 and standard deviation20. How big should n be so that the sampling distribution of612d9ae3240d8.jpg has standard deviation 1? (A)400 (B) 20 (C)Approximately 5 (D)Cannot be determined unless we know the population follows a normal distribution.

3. A researcher is interested in the size of the current balance of credit card holders. To estimate this, he obtains the size of the current balance of a random sample of 25 credit card holders. A 90% confidence interval for the mean current balance of credit card holders is found to be S662.72±$44.70. Which of the following would produce a contidence interval with a smaller margin of etror than this 90% confidence interval? (A)Obtain the balances of only five credit card holders rather than 25, because five are likely to be more uniform than 25. (B) Obtain the balances of 100 credit card holders rather than 25. (C)Computer a 99% confidence interval rather than a 90% confidence interval. The increase in confidence indicates that we have a better interval. (D)None of the above.

4. In a large city, the percent of total spending that households devote to housing is normally distributed, with mean u and standard deviation o= 9%. I select a simple random sample of four young households and determine the percent of their total spending that is devoted to housing. The four percentages are
 33%  39%  32%  36%
Based on these data, a 99% confidence interval forμis(A)35%±5.78%. (B)35%±8.82% (C)35%±11.59% (D)35%±23.18% 
5. An engineer designs an improved light bulb. The previous design had an average lifetime of 1200 hours. The new bulb has a lifetime of 1201 hours, using a sample of 2000 bulbs. Although the difference is quite small, the cffect is statistically significant. The explanation is (A)that new designs typically have more variability than standard designs. (B) that the sample size is very large. (C)that the mean of 1200 is large. (D)all of the above.
6. The sum of squares for error is (A)1021.8 (B)14512.0 (C)20853.5(D)147144.0
7. In the ANOVA table, the test for the main cffect of DESIGN has a P.-value of .613. This indicates that (A)sales probably vary considerably for the different designs. (B) for about 61.3% of the samples, there was a difference in the effect of design. (C)for about 61.3% of the samples, there was no difference in the effect of design. (D)None of the above is true.
8. the numerical value of the F statistic for testing interaction is(A)0.59(B)1.70(C)3.40(D)202.33
9. Investing is a game for chance. Suppose there is a 40% chance that a risky stock investment will end up in a total loss of your investment. Because the rewards are so high, you decide to invest in four independent risky stocks. Find the probability that at least one of your investments becomes a total loss. (A).0256 (B).40 (C).9744 (D).8704
10. The on-line access computer service industry is growing at an extraordinary rate. Current estimates suggest that only 20% of the home-based computers have access to on-line services. This number is expected to grow quickly over the next five years. Suppose 25 people with home-based computers were randomly and independently sampled.Find the probability that fewer than half of those sampled currently have access to on-line service. (A) 1,000 (B) .998 (C)994 (D).999

11. A small company conducted a survey of its employees to determine their attitudes towards a buyout of the company's stock. The company's 1,040 employees were asked whether they favored a buyout and when they were planning to retire. The data is shown in the accompanying table:
  612d9cbfba2c0.jpgWhich of the following is true between events A and B? (A)A and B are independent of onc another. (B) A and B are mutually exclusive events. (C)A and B are complimentary events. (D)A and B are dependent events.
12. A beer dispensing machine flls thousands of 12 ounce beers each day. The machine is very consistent but the pours do vary a little bit. Assume it is known that the distributions of pours is normally distributed with a mean of 12.1 ounces and a standard deviation of .05 ounces. Customers get very angry when their beers have less than the advcrtised 12 ounces of beer in them. Find the proportion of the customers that could get angry because of this reason. (A).4772 (B).9772 (C).0.228 (D).5228 1.040
13. Recent market research reveals that an estimated 30% of U.S. households own one or more personal computers (PCs). Supposc that in a sample of 500 houscholds in a large high-tech community, 178 own PCs. What distribution is appropriate to use to estimate the binomial random variable x = the number of the 500 homes that own PC's? (A)normal (B) exponential (C)Poisson (D)uniform

14. we never conclude "Accept HO" in a test of hypothesis. This is because: (A) α is the probability of a Type I error. (B) The rejection region is not known. (C)The p-value is not small enough. (D)β= p (type II error) is not known.
15. The manufacturet of an over-the-counter pain reliever claims that its product brings pain relief to headache sufferers in less than 3.5 minutes, on average. To be ablc to make this claim in its television advertisements, the manufacturer was required to present statistical evidence in support of their claim. The manufacturer reported that for a random sample of 50 headache sufferers, the mean time to relief was 3.3 minutes and the standard deviation was 60 seconds. Find the rejection region for the test if testing at a= .05. (A)Reject H0 ifz＜-1.96. (B) Reject  H0 ifz＜-1.645. (C)Reject  H0 ifz＜-1.282. (D)Reject H0 ifz ＜ -1.96 or z ＞1.96.
16. Researchers have claimed that the average number of headaches during a semester of Statistics is 14. Statistics professors dispute that this claim vehemently. Statistic professors believe the av average is much more than this. They sample n = 13 students and find the sample mean is 16 and the sample standard deviation is 2.0. The test statistic for this test is: (A)t =3.6 (B) t=1.0 (C)1=1.8 (D)t =2.1

17. A study was commissioned to compare the housing costs of two growing cities. The goal of the study was to estimate the difference in the average housing costs (as measured in price per square foot) of the two cities. Two pilot samples were taken and yielded the following information: 612d9d7069d9e.jpg Use this information to determine how large the sample sizes must be in order to estimate the true difference in the average housing costs of the two cities to within $1.25. Assume that equal sample sizes are desired and a 95% reliability level is essential. (A)nl =n2 =76 (B) n1 =n2=84 (C)n1 =n2=119 (D)n1 =n2= 25

18. State the null hypothesis necessary to test whether the proportion of trsactions assigned to the cach of the five memory locations differ. (A)612d9e0d5b263.jpg (B) 612d9e2751814.jpg (C)612d9e506687f.jpg (D)612d9e6e3492d.jpg
19. Calculate the test statistic necessary to test whether the proportion of transactions assigned to each of the five memory locations differ. (A)x2=6.31 (B) x2 =142 (C)X2 =7.25 (D)x2 = 5.00

20. Find the rejection region necessary for testing whether the proportion of transactions assigned to each of the five memory locations differ. Use α = .05.(A) Reject H0 if x2＞1.145476 (B) Reject H0 if x2 ＞0.710721 (C)Reject H0 if x2 ＞ 11.0705 (D)Reject H0 if x2 ＞9.48773

申論題 (4)

(a) estimates ofσ, where σ = 1/(I+p), the elasticity of substitution between Iabor and capital. (10 points)

(b) estimates of ν , the degree of returns to scale. (10 points)
(a) Discuss briefly the economic reasoning behind these specifications of the three theories. (15 points)
(b) Evaluate the three hypotheses. (15 points)