Question

You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% of the population proportion.
b. Find the minimum sample size needed, using a prior study that found that 32% of U.S. adults have taken or planned to take a winter vacation in a recent year. (Source: Rasmussen Reports)

Accepted Answer

Step 1: Identify the formula for determining the minimum sample size for estimating a population proportion. The formula is: n = $$(Z^2 * p * (1 - p$$)) / $$E^2$$, where n is the sample size, Z is the z-score corresponding to the desired confidence level, p is the estimated population proportion, and E is the margin of error. Step 2: Determine the values for the variables in the formula. For a 95% confidence level, the z-score (Z) is approximately 1.96. The estimated population proportion (p) is 0.32, and the margin of error (E) is 0.05. Step 3: Substitute the values into the formula. This gives: n = $$(1.96^2 * 0.32 * (1 - 0.32$$)) / $$0.05^2. $$Step 4: Simplify the expression step by step. First, calculate $$1.96^2$$, then multiply it by 0.32 and (1 - 0.32), and finally divide by $$0.05^2. $$Step 5: Round the result up to the nearest whole number, as the sample size must be a whole number. This will give you the minimum sample size needed to estimate the population proportion with the specified accuracy and confidence level.

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Key Concepts

Confidence Interval

Sample Size Calculation

Margin of Error