I am in a statistics class at Michigan State and am having trouble completing this problem that was assigned for homework. If somebody could give me some advice/pointers I would greatly appreciate it.

A Gallup Poll finds that 30% of adults visited a casino in the past 12 months, and that 6% bet on college sports. These results come from a random sample of 1011 adults. For an SRS of size n=1011:

a) What is the probability that the sample proportion P-hat is between .28 and .32 if the population proportion is p=.30?

b)What is the probability that the sample proportion P-hat is between .04 and .08 if the population proportion is p=.06?

c) How does the probability that P-hat falls within (+or-)0.02 of the true p change as p gets closer to 0?

I thought I figured out a), however my answer did not match what was in the back of the book. The method I used was to find the standard deviation (which I computed as .01441) and to solve for each probability based on the Z-score equation (Z= (X-m)/SD). Once i had both probabilities from the z-scores, I subtracted the probability that it was below .28 from the probability that was below .32, leaving me with the probability that it was in the middle of the two. I came up with .7685, however the back of the book has the correct answer as .8354.

Any help is appreciated, thank you!