私立中原大學八十九學年度碩士班招生考試命題紙


所組別： 資訊管理學系 科目： 統計學 考試時間： 4月29日 第4節

1. A magazine states the following hypotheses about the mean age of its subscribers. 

H0：μ = 28
Ha：μ≠28

a. The population standard deviation is known at σ = 6 years and the sample size is 100. With α = .05, what is the probability of accepting H0 for μ equal to 26 and 29? (6%)
b. What is the power at μ = 26? (3%)
c. If the manager conducting the test will permit a .15 probability of making a Type II error when the true mean age is 29, what sample size should be selected ? (Assume σ = 6 and a .05 level of significance.) (6%)
2. Three top-of-the-line intermediate-size automobiles manufactured in the United States have been test-driven and compared on a variety of criteria by a well-known automotive magazine. In the area of gasoline mileage performance, five automobiles of each brand were each test-driven 500 miles; the miles per gallon data obtain are reported below.
Automobile A 19 21 20 19 21
B 19 20 22 21 23
C 24 26 23 25 27
　
a. Use the analysis of variance (ANOVA) procedure with α = .05 to determine whether there is a significant difference in the mean number of miles per gallon for the three types of automobiles. (6%)
b. Use the Kruskal-Wallis test with α = .05 to determine whether there is a significant difference in the mean number of miles per gallon for the three types of automobiles. (6%)
　
3. A hotel has 120 rooms. In the spring months, hotel room occupancy is approximately 75%. Use the normal approximation to the binomial distribution to calculate the probability that 80 or fewer rooms are occupied on a given day? (6%)
4. A company has recorded data on the daily demand for their product (Y in thousands of units) and their unit price (X in hundreds of dollars). A sample of 10 days demand and associated price resulted in the following data.
∑X =  75　　　∑X2 = 637
∑Y =110　　　∑Y2 = 1,540
∑XY = 676
a. Develop the least squares estimated regression line. (3%)
b. Compute the coefficient of determination. (3%)
c. Perform an F test and determine whether or not there is a significant relationship between demand and unit price. Let α = 0.05. (3%)
d. Perform a t test to determine whether the slope is significantly different from zero. Let α = 0.05. (3%)
e. Develop a 95% confidence interval for the demand if expected price is $300. (3%)
　
5. A portion of the Minitab computer output follows.
The regression equation is Y = 14.4 – 8.69 X1 + 13.52 X2
Predictor Coef Stdev t-ratio
Constant 14.448 8.191 1.76
X1 　 1.555 　
X2 13.157 2.085 　

S = 3.773　　R-sq =　(1)　%　R-sq (adj) =　(2)　%

Analysis of Variance
SOURCE DF SS MS F
Regression 2 　 　 　(3)　
Error 　 71.17 　 　
Total 7 720.0 　 　

a. Calculate the missing entries (1)(2)(3) in this output. (6%)
b. Use the t test and α = .05 to test H0：β2 = 0. (4%)
6. A Bayesian approach can be used to revise probabilities that a prospect field will produce oil. In one case, geological assessment indicates a 25% chance that the field will produce oil. Further, there is an 80% chance that a particular well will strike oil given that oil is present in the prospect field.
a. Suppose that one well is drilled on the field and it comes up dry. What is the probability that the prospect field will produce oil? (3%)
b. If two wells come up dry, what is the probability that the field will produce oil? (3%)
c. The oil company would like to keep looking as long as the chances of finding oil are greater than 1%. How many dry wells must be drilled before the field will be abandoned? (4%)
7. Two faculty members ( X and Y ) ranked five candidates for scholarships. The rankings are shown below.
Candidate Peter Nancy Michael Mary Judy
Rank By X 5 2 1 3 4
Rank By Y 1 2 3 5 4

a. Compute the Spearman rank-correlation (5%)
b. Test it for significance. Let α = 0.05. (5%)
　
8. A survey firm conducts door-to-door surveys on a variety of issues. Some individuals cooperate with the interviewer and complete the interview questionnaire and others do not. The following data are available
Respondents Sample Size Number Cooperating
Men 200 110
Women 300 210
　
a. Using α = 0.05, test the hypothesis that the response rate is the same for both men and woman. (5%)
b. Compute the 95% confidence interval for the difference between the proportions of men and woman who cooperate with the survey. (5%)
　
9. A professor believes that the final examination scores in statistics are normally distributed. A sample of 40 final scores has been taken. You are given the sample below. The mean of the scores is 83.1, and the standard deviation is 10.43.
56 63 65 68 72 72 73 75 77 78
78 79 80 80 80 80 80 80 81 81
82 84 84 86 86 87 88 90 90 92
92 93 93 94 95 96 97 98 100 100
　
a. Compute the test statistic for the goodness of fit test.(4%)
b. The hypothesis is to be tested at the 1% level. What is the critical value from the table for the test? (4%)
c. What do you conclude about the distribution of final examination scores? (4%)




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