102年國立政治大學企管所(甲組)統計學

A statistics professor classifies his students according to their grade point average (GPA) and their class rank. GPA is on a 0.0-4.0 scale, and class rank is defined as the under class (freshmen and sophomores) and the upper class (juniors and seniors). One student is selected at random.

(1) If the student selected is in the upper class, what is the probability that her GPA is between 2.0 and 3.0? 
(2) If the GPA of the student selected is over 3.0, what is the probability that the student is in the lower class? 
(3) Are being in the upper class and having a GPA over 3.0 related? Explain. 
(4) If 10 students are selected randomly, what is the probability that at least one has GPA over 3.0?

How much money does a typical family of four spend at McDonald’s restaurants per visit? The amount is a normally distributed random variable whose mean is $16.40 and whose
standard deviation is $2.75.

(1) Find the probability that a family of four spends less than $15.

(2) What is the amount below which only 10% of families of four spend at McDonald’s?

(3) What is the probability that the mean amount of a random sample of 64 family of four spends less than $15?

In a number of pharmaceutical studies volunteers who take placebos (but are told they have taken a cold remedy) report the following side effects: 
Headache(1) 6% Drowsiness (2) 8% Stomach upset(3) 4% No side effect (4) 82%
A random sample of 250 people who were given a placebo (but who thought they had taken anti-inflammatory) reported whether they had experiences each of the side effects. The frequency distribution of these responses were as follows: Headache 19, Drowsiness 23, Stomach upset 14 and No side effect 193. Do these data provide enough evidence at 5% significance level to infer that the reported side effects of the placebo for an anti-inflammatory differ from that of a cold remedy (taking placebos)?

The joint probability distribution of variables X and Y is shown in the table below, where X is the number of tennis racquets and Y is the number of golf clubs sold daily in a small sports store.

(1) Find the marginal probability distributions of X and Y.
(2) Find the probability distribution of the random variable X + Y . 
(3) Find COV (X , Y) .

The following density function describes the random variable X.
f (x) = x / 25   0 < x < 5 ;  (10 − x) / 25   5 < x < 10 .
(1) Find the probability that X lies between 4 and 8. 
(2) Find E(X ) and Var(X ) .

An engineering student who is about to graduate decides to survey various firms in Silicon Valley to see which offered the best chance for early promotion and career advancement. He surveyed 30 small firms (size level is based on gross revenues), 30medium-size firms, and 30 large firms and determined how much time must elapse before an average engineer can receive a promotion.
(1) Complete the ANOVA table. 
(2) Can the engineering student conclude at 5% significance level that speed of promotion
varies between the three sizes of engineering firms? 
(3) If difference exist, which of the following is true? Use LSD method. 

1. Small firms differ from the other two.
2. Medium-size firms differ from the other two.
3. Large firms differ from the other two.
4. All three firms differ from one another.
5. Small firms differ from large firms.

Detergent manufacturers frequently make claims about the effectiveness of their products.
A consumer protection service decided to test the five best-selling brands of detergent,where each manufacturers the its product produces the “whitest whites” in all water temperatures. The experiment was conducted in the following way. One hundred fifty white sheets were washed in each brand-10 with cold water, 10 with warm water, and 10 with hot water. After washing, the “whiteness” scores for each sheet were measured with laser equipment.
(1) Complete the ANOVA table. 
(2) Can we infer at 5% significance level that there are differences in whiteness scores between the five detergents, 
(3) Can we infer at 5% significance level that there are differences in three water temperatures 
(4) Can we infer at 5% significance level that there are interaction between detergents and temperatures.
(5) For the three tests in (2)~(4), which one should be performed first? Why? What would you do if this test is found to be significant?