1. Suppose 7% of construction workers have been exposed to asbestos. We randomly sample 4 construction workers (assume they are independent). Use the Binomial probability function, with n = 4 and p = .07, to find the probability that:

a. exactly two of the workers have been exposed to asbestos

b. at least one of the workers has been exposed to asbestos (use the complement law)

2. Suppose 25% of rats have recessive traits. We take a random sample of 16 rats. Use the Binomial table to answer the following.

a. Find the probability that exactly 5 rats have recessive traits.

b. Find the probability that no more than 9 rats have recessive traits.

c. Find the probability that at least 5 rats have recessive traits.

d. Find the probability that between 4 and 10 rats, inclusive, have recessive traits.

e. Suppose we observe that 2 of the 16 rats have recessive traits. Would you consider this an unusually low number? Calculate a probability to justify your answer.

3. Suppose 55% of babies born in a certain country are female. We take a random sample of 20 babies from this country. Use the Binomial table to answer the following.

a. Find the probability that exactly 15 babies are female.

b. Find the probability that no more than 12 babies are female.

c. Find the probability that at least 4 babies are female.

d. Suppose we observe that 17 of the 20 babies are female. Would you consider this an unusually high number? Calculate a probability to justify your answer.

4. In a small pharmacy, suppose there is an average of 2 prescriptions of a certain drug per week. Use Poisson tables to answer the following.

a. Find the probability that there are exactly 2 prescriptions for this drug in a given week.

b. Find the probability that there are no more than 5 prescriptions for this drug in a given week.

c. Find the probability that there is at least 1 prescription for this drug in a given week.

d. Suppose we observe 7 prescriptions for this drug in a given week. Would you consider this an unusually high number? Calculate a probability to justify your answer.

5. Let Z represent a standard Normal random variable.

a. Find P( Z < 1.96 )

b. Find P( Z < -3.03 )

c. Find P( Z > 0.67 )

d. Find P( -1.37 < Z < 2.60 )

e. P(Z <10)

f. Find the point C such that P( Z < C ) = .2

g. Find the point C such that P( Z > C ) = .33

6. Suppose reaction time in some population is Normally distributed with mean 470 and standard deviation 40 (both measured in milliseconds). We randomly select a person from this population, and measure the reaction time.

a. Find the probability the reaction time is 500 or less.

b. Find the probability the reaction time is at least 400.

c. Find the probability the reaction time is between 450 and 550.

d. Find the 96th percentile of the distribution of reaction time (at least approximately). That is, find the value such that about 96% of the distribution is below this point.

7. Suppose women with anorexia have a daily caloric intake that is Normally distributed with mean 1385 and standard deviation 65. Further, suppose women without anorexia have a daily caloric intake that is Normally distributed with mean 1720 and standard deviation 110. We use a single day's caloric intake as a screening test for anorexia. We decide a value under 1500 is a positive screen, so a value over 1500 is a negative screen.
a. Briefly explain why a relatively low value of this variable is a positive screen.

b. Calculate the sensitivity of the test. This is the probability of having a positive screen, given that the person has the disease, so use the distribution for women with anorexia.

c. Calculate the specificity of the test. This is the probability of having a negative screen, given that the person does not have the disease, so use the distribution for women without anorexia.

d. Suppose we changed the cutoff point from 1500 to 1600. Would the sensitivity increase or decrease? You don't have to make any calculations if you can answer this without doing so.

e. Suppose we changed the cutoff point from 1500 to 1600. Would the specificity increase or decrease? You don't have to make any calculations if you can answer this without doing so.

f. Briefly describe how we could decide on the "best" cutoff point.

8. As in problem #3, suppose 55% of babies born in a certain country are female. Now, we take a random sample of 200 babies from this country. Use the Normal approximation to the Binomial to find the probability that at least 100 babies are female.