98年國立臺北大學企業管理學系統計學線上測驗

1.: A confidence interval was used to estimate the proportion of statistics
students who are female. A random sample of 72 statistics students
generated the following confidence interval: (.438, .642). Using the
information above, what sample size would be necessary if we wanted to
estimate the true proportion to within 3% using 95% reliability? Please
explain your answer with calculating process. (15%)
(A): 1061
(B): 1068
(C): 1110
(D): 1025

2.

A certain HMO is attempting to show the benefits of managed care to an
insurance company. The HMO believes that certain types of doctors are
more cost-effective than others. One theory is that certification level is an
important factor in measuring the cost-effectiveness of physicians. To
investigate this, the HMO obtained independent random samples of 20
physicians from each of the three certification levels-Board certified (C);
Uncertified, board eligible (E); and Uncertified, board ineligible (I)-and
recorded the total per member per month charges for each (a total of 60
physicians). The results of the ANOVA are summarized in the following
table. Take α = 0.01 .
a. What kind of design was it conducted? (4%)
b. Complete the ANOVA table. (6%)

c. Interpret the p-value of the ANOVA F-test. (4%)
A) The means of the total per member per month charges for the three
      groups of physicians differ at α = .01 .
B) The model is not statistically useful (at α = .01 ) for prediction
      purposes.
C) The variances of the total per number per month charges for the
      three groups of physicians differ at α = .01 .
D) The means of the total per member per month charges for the three
      groups of physicians are equal at α = .01 .
d. What assumptions are necessary for the validity of the F statistic? (6%)

3.

a. State the assumptions necessary for predicting the monthly sales based
on the linear relationship with the months on the job. (5%)
b. Fit the least squares line on the scatter plot of (months on the job,
monthly sales). (5%)
c. Please predict the monthly sales for two years on the job. (5%)

4.

5.: A chemical company manufactures “long-life” batteries. Extensive testing
has shown that the chance that a battery of type will fail within 3 years is
20%. The company sells the batteries in packages of 4. You are about to
buy a package of these batteries for your portable CD player, which
requires 4 batteries. The CD player will fail to operate if at least one of the
batteries fails. In this case, you will have to return to the store to buy new
batteries. What is the probability that you will NOT have to return to the
store within 3 years? (10%)

6.: a. State a version of the Central Limit theorem that applies to
independent, identically distributed random variables. (8%)
b. Let X have a binomial (n = 45, p = 1/3) distribution. Use the central
Limit Theorem to approximate P(X ≤ 17) . (8%)
c. Given an example of a population with finite variance to which the
Central Limit Theorem does not apply. Explain why it does not apply.
(8%)

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