Based on a study conducted by the Gallup organization, 76% of Americans believe that the state of moral values in the United States is getting worse. Describe the sampling distribution of the sample proportion for samples of size

(a) n = 10,

(b) n = 25,

(c) n = 60.

Approach Describing a distribution means finding its shape, center, and spread. The actual sampling distribution of the sample proportion would be the distribution of all possible sample proportions of size n = 10. It is virtually impossible to find all possible samples of size n = 10 from the population of Americans. To get a sense as to the shape, center, and spread of the sampling distribution of the sample proportion, we will use StatCrunch’s Bernoulli Random Data command with event probability 0.76 to simulate obtaining 2000 samples of size n = 10 by randomly generating 2000 rows of responses over 10 columns. Each row consists of a 0 or 1, with 0 representing a failure (individual does not believe the state of moral values is getting worse) and 1 representing a success (individual believes the state of moral values is getting worse). For each of the 2000 samples (the 2000 rows), determine the mean number of successes (the sample proportion). Draw a histogram of the 2000 sample proportions to gauge the shape of the distribution of the sample proportions. Determine the mean of the 2000 sample proportions to approximate the mean of the sampling distribution. Determine the standard deviation of the 2000 sample proportions to approximate the standard deviation of the sampling distribution.

Repeat this process for samples of size n = 25 and n = 60.