Question

A researcher studying public opinion of proposed Social Security changes obtains a simple random sample of 30 adult Americans and asks them whether or not they support the proposed changes. To say that the distribution of the sample proportion of adults who respond yes, is approximately normal, how many more adult Americans does the researcher need to sample in the following cases?

A) 10% of all adult Americans support the changes
B) 15% of all adult Americans support the changes
C) The researcher must ask 82 more American adults.
D) The researcher must ask more American adults.

Answer 1

To determine if the distribution of the sample proportion of adults who respond yes is approximately normal, we need to consider the conditions for a binomial distribution and use the rule of thumb for sample size. For case A, 10 additional adult Americans would need to be sampled. For case B, 15 additional adult Americans would be needed. For case C, we need to determine the sample proportion and check if the conditions for a normal distribution are met.

Step-by-step explanation:

To determine if the distribution of the sample proportion of adults who respond yes is approximately normal, we need to consider the conditions for a binomial distribution:

1. Each observation is independent.
2. There are only two possible outcomes: yes or no.
3. The probability of success (yes) is constant for each observation.
4. The number of observations is fixed.

Based on the information given, we can assume these conditions are met. To determine the sample size needed for an approximately normal distribution, we can use the rule of thumb that states that if both np and n(1-p) are greater than or equal to 10, then the distribution can be approximated as normal.

A) If 10% of all adult Americans support the changes, we can estimate the proportion as 0.10. Let p represent the proportion of support in the population. The sample size needed is n = np / (p(1-p)), where n is the number of additional adults needed to sample, and p is the estimated proportion of support in the population. Substituting the values, we get n = (0.10 * n) / (0.10 * 0.90). Simplifying, we find n = 10 additional adult Americans.

B) If 15% of all adult Americans support the changes, using the same formula, we get n = 15 additional adult Americans.

C) If the researcher must ask 82 more American adults, we can find the sample proportion by dividing the number of adults who respond yes by the total sample size. We can then use this sample proportion to determine if the conditions for a normal distribution are met using the rule of thumb. If they are, then the distribution of the sample proportion can be approximately normal.

D) If the researcher must ask more American adults, we cannot determine an exact number without additional information.