課程名稱︰統計學一上 課程性質︰必修 課程教師︰王衍智 開課學院：管理學院 開課系所：財務金融學系 考試日期（年月日）︰102.01.02 考試時限（分鐘）：100 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Statistics I mock examination Instructor: Yanzhi Wang Time: 9:10-10:50. Note: This is a closed-book test. Calculator and dictionary are allowed. Pencil is NOT allowed. Please mark your answers on blank around each question on the test paper. To simplify your answer, you can show answers with 3 digits after the decimal point. Part I: multiple choices question (30%, 3% for each). 1. Suppose that A and B are two independent events. The probability that event A occurs is 0.3 (i.e., P(A) = 0.3), and that B occurs is P(B) = 0.2. What is the probability that A does not occur and B also does not occur? A) 0.06 B) 0.20 C) 0.30 D) 0.56 E) 0.60 2. The weights of extra-large eggs have a Normal distribution with a mean of 1 oz and a standard deviation of 0.1 oz. What is the sampling distribution of the sample mean weight of the eggs in a randomly selected carton of a dozen eggs? A) N(1, 0.029) B) N(12, 0.1) C) N(1, 0.1) D) N(12, 1) E) N (1, 1) 3. The time college students spend on the internet follows a Normal distribution. At Johnson University, the mean time is 5 hrs with a standard deviation of 1.2 hrs. What is the probability that the average time 100 random students on campus will spend more than 5 hours on the internet? A) 0.5 B) 1 C) 0 D) 0.2 E) 0.1 4. During the summer months, the prices of nonsmoking rooms with a king-size bed in hotels in a certain area are roughly Normally distributed with a mean of $132 and a standard deviation of $29. What percentage of nonsmoking rooms with a king-size bed cost more than $150? A) 18.94% B) 26.76% C) 73.24% D) 81.06% E) within ± 0.005 of any of the above is incorrect. 5. A nationally distributed college newspaper conducts a survey among students nationwide every year. This year, responses from a simple random sample of 204 college students to the question “About how many CDs do you own?” ─ resulted in a sample mean of x = 72.8. Based on data from previous years, the editors of the newspaper will assume that σ= 7.2. Which one of following statement is NOT true? A) The sample mean lie in the 95% confidence interval. B) The population mean may not lie in the 95% confidence interval. C) If we were to use a 92% confidence level, the confidence interval from the same data produce an interval wider than the 95% confidence interval. D) With a smaller sample size, all other things being the same, the 95% confidence interval would be wider. E) We can use normal distribution to test the significance. 6. A researcher plans to conduct a test of hypotheses at the 1% significance level. She designs her study to have a power of 0.90 at a particular alternative value of the parameter of interest. What is the probability that the researcher will commit a Type I error? A) 0.01 B) 0.90 C) 0.10 D) 0.99 7. Which of the following statements is true? A) If the sample size decrease from 150 to 75, the margin of error would have been larger. B) If we request high C for confidence interval, the margin of error would have been smaller. C) If we request high α on significance level, the margin of error would have been smaller. D) The result that describes why we are able to calculate a confidence interval ─ for the mean X, even when the population distribution is non-Normal, is called the weak law of large number. E) Two of above are correct. 8. Which of following statements about the point estimator is correct? A) An unbiased estimator must be a most efficient estimator. B) An unbiased estimator must be a most consistent estimator. C) A biased estimator could be a most efficient estimator. D) A biased estimator cannot be a consistent estimator. E) When sample size approaches infinite and the power of a statistic is approaching one, then we will say this statistic is unbiased. 9. Assume that sample data, based on two independent samples with both size 25, ─ ─ give us x1 = 505, x2 = 515, s1 = 23, and s2 = 28. Determine which of the following statements is NOT true. A) Based on the confidence interval, we can conclude at the 5% significance level that there is no difference between the two population means,μ2 and μ1. B) The margin of error for the difference between the two sample means would be smaller if we were to take larger samples. C) If we were to use the unpooled t-test with the more accurate approximation for the degrees of freedom, the degrees of freedom would be 51. D) If a 99% confidence interval were calculated instead of the 95% interval, it would include more values for the difference between the two population means. E) Two variables are statistically indifferent under 5% significance level (F =2.3). 24,24,α=0.025 10. A newspaper is conducting a statewide survey concerning the race for governor. The newspaper will take a simple random sample of n registered voters and determine X = the number of voters that will vote for the Democratic candidate. Is there evidence that a clear majority of the population will vote for the Democratic candidate? To answer this, they will test the hypotheses H0: p = 0.5 versus Ha: p > 0.50. If n = 1200 and X = 640, what is the P-value around for this hypothesis test? A) Less than 0.0002 B) 0.0105 C) 0.0330 D) 0.2326 E) within ± 0.005 of any of the above is incorrect. Part II: Calculations (40%, 10% for each). 11. You flip a coin 400 times to test if it is a fail coin (it is fail if chance of head is 0.5). Among 400 flips, you get 160 heads and 240 tails. Please test if this coin is a fail one with two-tail 5% significance level. 12. A joint probability chart is shown below: Y=1 Y=2 Y=3 Y=4 X=1 0.1 0 0 0 X=2 0.2 0.05 0.15 0 X=3 0.15 0.05 0.1 0.2 Please find E(X|Y) 13. Following previous question, please find E(X*Y^0.5|Y) 14. A firm claims that the duration of battery is 30 hours; we assume the standard deviation σ is known and equals 4. If we random sample n batteries under true battery duration is 28.5 hours. What is the n should be if we request the power of test equal to 0.9 (significance level α=0.05 under one-tail test)? 15. A quality controller tests the average of beverages, which are with unknown standard deviations. Four random samples are 844, 847, 845 and 844 grams. Please fine the 95% conference interval. Part III: Proofs (20%) 16. Please prove cov(x,y)^2≦var(x)var(y) 17. Given X~(μ, 1). Please show whether W is a consistent estimator on μ^0.5, N given W=√Z , Z=( ΣXi)/N i=1 標準常態分配表: http://imgur.com/uLiJoZD T分配臨界值表: http://imgur.com/vN29lGc --

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