Question

a social scientist wishes to conduct a survey. she plans to ask a yes/no question to a random sample from the u.s. adult population. one proposal is to select 100 people; another proposal is to select 900 people. which of the following is true regarding the sample proportion p^ of \yes\ responses? a. the sample proportion from the sample of 900 is more likely to be close to the true population proportion, p. b. the sample proportion from sample of 100 is more likely to be close to the true population proportion, p. c. the sample proportion in either proposal is equally likely to be close to the true population proportion, p, since the sampling is random. question 3 a political polling agency wants to take a random sample of registered voters and ask whether or not they will vote for a certain candidate. one plan is to select 400 voters, another plan is to select 1,600 voters. if the study were conducted repeatedly (selecting different samples of people each time), which one of the following would be true regarding the resulting sample proportions of \yes\ responses? a. different sample proportions would result each time, but for either sample size, they would be centered (have their mean) at the true population proportion. b. for either sample size, using the same size each time, as long as the samples are drawn with replacement, they would be centered (have a mean) at 0. c. different sample proportions would result each time, but for sample size 400 they would be centered (have their mean) at the true population proportion, whereas for sample size 1,600 they would not. d. different sample proportions would result each time, but for sample size 1,600 they would be centered (have their mean) at the true population proportion, whereas for sample size 400 they would not.

Explanation:

Step1: Recall sampling - proportion concept

Larger sample sizes reduce sampling variability.

Step2: Analyze first - question options

A larger sample (900 people) is more likely to have a sample proportion close to the true population proportion due to less sampling error. So, for the first question, option A is correct.

Step3: Recall properties of sampling distribution of sample proportion

The mean of the sampling distribution of the sample proportion $\hat{p}$ is equal to the population proportion $p$ for any sample size when sampling is random.

Step4: Analyze second - question options

For repeated sampling with different sample sizes, the sample proportions will vary, but the mean of the sampling distribution of $\hat{p}$ is the true population proportion $p$ for both sample sizes. So, for the second question, option A is correct.

Answer:

Question 2: A. The sample proportion from the sample of 900 is more likely to be close to the true population proportion, p.
Question 3: A. Different sample proportions would result each time, but for either sample size, they would be centered (have their mean) at the true population proportion.