111年 - 111 國立臺北大學_碩士班一般入學考試試題_統計學系：統計學#147168

a. (7%) What is the probability that a 55-year-old, disease-free male will develop colon cancer over the next 5 years?

b. (8%) Suppose there is a cohort of 1000 50-year-old men who have never had cancer. How many colon cancers would be expected to develop in this cohort over a 15-year period?

a. (7%) If the binomial distribution holds, what is the probability distribution of the number of affected people in rooms with 2 occupants? That is, what is the probability of finding zero affected people? One affected person? Two affected people?

b. (10%) One useful summary measure of geographical clustering is the number of rooms with 2 or more affected occupants. If the binomial distribution holds, what is the expected number of rooms with 2 or more affected occupants over the entire nursing home?

c. (3%) Compare the observed number of rooms with 2 or more affected occupants with the expected number of rooms. Does this comparison give any evidence that clustering of disease occurs within rooms?

Table 2

Number of affected people and total number of people in a room for an outbreak of acute gastroenteritis in a nursing home in Baltimore, Maryland

| People in room | Total number of rooms | Number of rooms with | | | | |
| --- | --- | --- | --- | --- | --- | --- |
| | | 0 affected people | 1 affected person | 2 affected people | 3 affected people | 4 affected people |
| 2 | 13 | 5 | 4 | 4 | 0 | 0 |
| 3 | 4 | 1 | 2 | 0 | 1 | 0 |
| 4 | 15 | 2 | 4 | 3 | 5 | 1 |

a. (10%) If the probability density function of the random variable \( X \) is \( f(x) \), find \( P(X < t) \) for all possible values of \( t \).

b. (5%) For all possible values of \( t \), find \( P(|X| < t) \).

a. (5%) If 9 students are taken as a simple random sample, what is the probability that the mean score will be between 67 and 71.5 points?

b. (5%) How many students must be selected so that the margin of error for the \( 95\% \) confidence interval of the mean score is within 5 points?

a. (5%) For a male student with IQ 100 and SC score 50 and his AP score is 4. What is the residual?

b. (5%) Conduct a test to determine if IQ score is an important predictor for AP score.

c. (5%) With confidence coefficient .95, how does AP score change for one unit increases in IQ score while holding SC score and SEX constant?

d. (5%) What is the test statistics (numerical value) for testing if NONE of the predictors are significant?

a. (5%) Conduct a test to determine if men are more willing to spend money on luxury cars than women.

b. (5%) Construct a \( 95\% \) confidence interval for difference in population proportions between men and women who are willing to spend money on luxury cars.

a. (5%) The population standard deviation is assumed known 6 years and the sample size is 100. With a significant level .05, what is the probability of NOT rejecting \( H_{0} \) for \( \mu \) equal to 26?

b. (5%) What is the power at \( \mu = 26 \)?