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 2Number 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?

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