Congress You wish to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.
b. Find the minimum sample size needed, using a prior survey that found that 21% of likely U.S. voters think Congress is doing a good or excellent job. (Source: Rasmussen Reports)

검증된 단계별 안내

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 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 0.21 (from the prior survey), and the margin of error (E) is 0.04 (4%).

Step 3: Substitute the values into the formula. This gives: n = (1.96^2 * 0.21 * (1 - 0.21)) / 0.04^2.

Step 4: Simplify the numerator of the formula. Calculate 1.96^2, then multiply it by 0.21 and (1 - 0.21), which is 0.79.

Step 5: Simplify the denominator of the formula. Calculate 0.04^2, then divide the simplified numerator by the simplified denominator to find the minimum sample size. Round up to the nearest whole number, as sample size must be an integer.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Population Proportion

Population proportion refers to the fraction of a population that exhibits a certain characteristic. In this context, it is the percentage of likely U.S. voters who believe Congress is performing well. Understanding this concept is crucial for estimating how representative a sample will be of the entire population.

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to ensure that the results of a survey or experiment are statistically valid. It takes into account the desired confidence level, margin of error, and the estimated population proportion, which in this case is 21%.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. In this scenario, a 95% confidence level means that if the survey were repeated multiple times, 95% of the calculated intervals would contain the true population proportion of voters' opinions on Congress.