Question

question 4 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 one of the following is true regarding the standard deviation of the sampling distribution of the sample proportion, p ^, of \yes\ responses? a. the standard deviation of the sampling distribution will be 9 times smaller with sample size 100. b. the standard deviation of the sampling distribution will be 9 times larger with sample size 100. c. the standard deviation of the sampling distribution will be 3 times smaller with sample size 100. d. the standard deviation of the sampling distribution will be 3 times larger with sample size 100. e. the standard deviation of the sampling distribution will be the same for both sample sizes. question 5 which of the following best describes the sampling distribution of a statistic? a. a distribution of a single statistic from repeated random samples of the same size, from the same population. b. a normal curve, for which probabilities are obtained by standardizing. c. a distribution of all parameters from the population that is to be randomly sampled. d. a distribution of all possible summary statistics from a single random sample, from the same population. e. the mechanism that determines whether the random sampling was effective.

Explanation:

Step1: Recall standard - deviation formula for sample proportion

The standard deviation of the sampling distribution of the sample proportion $\hat{p}$ is given by $\sigma_{\hat{p}}=\sqrt{\frac{p(1 - p)}{n}}$, where $p$ is the population proportion and $n$ is the sample size.

Step2: Compare standard - deviations for different sample sizes

Let $n_1 = 100$ and $n_2=900$. Then $\sigma_{\hat{p}_1}=\sqrt{\frac{p(1 - p)}{n_1}}$ and $\sigma_{\hat{p}_2}=\sqrt{\frac{p(1 - p)}{n_2}}$. The ratio $\frac{\sigma_{\hat{p}_1}}{\sigma_{\hat{p}_2}}=\sqrt{\frac{n_2}{n_1}}$. Substituting $n_1 = 100$ and $n_2 = 900$, we get $\frac{\sigma_{\hat{p}_1}}{\sigma_{\hat{p}_2}}=\sqrt{\frac{900}{100}} = 3$. So $\sigma_{\hat{p}_1}=3\sigma_{\hat{p}_2}$, which means the standard deviation of the sampling distribution will be 3 times larger with sample size 100.

Step3: Recall the definition of sampling distribution of a statistic

The sampling distribution of a statistic is the distribution of a single statistic from repeated random samples of the same size, from the same population.

Answer:

Question 4: D. The standard deviation of the sampling distribution will be 3 times larger with sample size 100.
Question 5: A. A distribution of a single statistic from repeated random samples of the same size, from the same population.