Question

A researcher wishes to​ estimate, with 99​% confidence, the population proportion of adults who think Congress is doing a good or excellent job. Her estimate must be accurate within 2​% of the true proportion.

​(a) No preliminary estimate is available. Find the minimum sample size needed.

​(b) Find the minimum sample size​ needed, using a prior study that found that 18​% of the respondents said they think Congress is doing a good or excellent job.

​(c) Compare the results from parts​ (a) and​ (b).

Answer 1

Here , the margin of error is E=0.02 ,

(a) Since the no estimate of the population proportion p is provided, we use the estimate p = 0.5.

The critical value for the significance level α=0.01 is z_c = 2.58. The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:

Therefore, the sample size needed to satisfy the condition , and it must be an integer number, we conclude that the minimum required sample size is n = 4147.

(b) The provided estimate population proportion p is p = 0.18.

The critical value for the significance level α=0.01 is z_c = 2.58. The following formula is used to compute the minimum sample size required to estimate the population proportion p within the required margin of error:

Therefore, the sample size needed to satisfy the condition , and it must be an integer number, we conclude that the minimum required sample size is n = 2449.