Researchers would like to estimate the proportion of adults who have dined out in the last week. They will report the sample proportion from a simple random sample (SRS) of adults as a point estimate for the population proportion. Which of the following statements must be true about the sample proportion as a point estimate for the population proportion? (A) If researchers select an SRS of size n = 200, the estimator will have more variability than if they select an SRS of size n = 400. (B) If researchers select an SRS of size n = 200, the estimator will have less variability than if they select an SRS of size n = 400. (C) If researchers select an SRS of size n = 200, the estimator will be biased, but if they select an SRS of size n = 400, the estimator will be unbiased. (D) If researchers select an SRS of size n = 200, the estimator will be unbiased, but if they select an SRS of size n = 400, the estimator will be biased. (E) Whether the researchers select an SRS of size n = 200 or n = 400, the estimator will be biased since the only way to have an unbiased estimator is to conduct a census.
Question
Researchers would like to estimate the proportion of adults who have dined out in the last week. They will report the sample proportion from a simple random sample (SRS) of adults as a point estimate for the population proportion. Which of the following statements must be true about the sample proportion as a point estimate for the population proportion? (A) If researchers select an SRS of size n = 200, the estimator will have more variability than if they select an SRS of size n = 400. (B) If researchers select an SRS of size n = 200, the estimator will have less variability than if they select an SRS of size n = 400. (C) If researchers select an SRS of size n = 200, the estimator will be biased, but if they select an SRS of size n = 400, the estimator will be unbiased. (D) If researchers select an SRS of size n = 200, the estimator will be unbiased, but if they select an SRS of size n = 400, the estimator will be biased. (E) Whether the researchers select an SRS of size n = 200 or n = 400, the estimator will be biased since the only way to have an unbiased estimator is to conduct a census.
Solution
The correct answer is (A) If researchers select an SRS of size n = 200, the estimator will have more variability than if they select an SRS of size n = 400.
Here's why:
In statistics, the variability of an estimator (in this case, the sample proportion) decreases as the sample size increases. This is because a larger sample size gives us more information about the population, which reduces the uncertainty (or variability) of our estimate. Therefore, an SRS of size n = 400 will have less variability than an SRS of size n = 200.
The other statements are not necessarily true. The bias of an estimator does not depend on the sample size, but on how the sample is selected. An SRS, regardless of its size, is designed to be unbiased as long as every member of the population has an equal chance of being selected. Therefore, both an SRS of size n = 200 and an SRS of size n = 400 can be unbiased, contradicting statements (C), (D), and (E).
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