In a certain population 27% of the residents are overweight. Choose three residents at random. Let X = the number of overweight residents in the sample.

USING METHODS FROM THIS WEEK ABOUT THE BINOMIAL DISTRIBUTION:

a) Find the probability all three residents in the sample are overweight. P(X=3) = ? Would this be unusual?

b) Find the probability none of the three residents in the sample are overweight. P(X=0)= ? Would this be unusual?

c) Use your answers above to fill in this table. You may find it helpful to write your answers in this table.

d) Graph (paper and pencil are great) the probability distribution of X (X on x-axis and probability on the y-axis). Even if you can't show the graph in your hw, do it on scratch paper

e) Find the E(X) = and Var (x) = ² using what you learned about a binomial mean and variance.

f) Find P(more than 1 out of 3 is overweight).

g) Find P(1 or more is overweight). Explain the difference between f and g.

70 percent of patients in a hospital were satisfied with their hospital stay. 10 patients are selected at random for a satisfaction survey.

a) Find the probability that six or fewer of patients were satisfied with their hospital stay.

b) Find the probability that four or more were dissatisfied.

c) What is the relationship between these two questions/events?

About 22% of the population takes multivitamins. Our study has 10,000 participants from this population. In our study we compute phat=p= proportion in sample who take multivitamins.

a) Compute E(X)= and Var(X)=², and explain them in words.

b) How is phat distributed (shape, mean, standard deviation . . .)?

c) If you found that phat = 0.24, would you be surprised with how large this value is? (Compute the P(phat>0.24), and comment on whether the probability is "small," which means it is an unusual event.) This is foreshadowing of p-values.

Low values of HDL (high-density lipoprotein) have been found to be associated with increased risk of heart disease. Assume that HDL levels in a population of women (ages 20 years or more) are approximately normally distributed with mean mu= 55 mg/dL and standard deviation sigma=15.5 mg/dL.

a) Find the percent of women who are considered to have a protective level of HDL, which is 60 mg/dL or greater.

b) A patient's HDL level is higher than 75% of the values in this population. Find that patient's HDL level.