Fast Food You wish to estimate, with 90% confidence, the population proportion of U.S. families who eat fast food at least once per week. Your estimate must be accurate within 3% of the population proportion.
a. No preliminary estimate is available. Find the minimum sample size needed.

검증된 단계별 안내

Step 1: Recall the formula for determining the minimum sample size for estimating a population proportion when no preliminary estimate is available: n = (Z^2 * 0.25) / E^2. Here, Z is the critical value corresponding to the desired confidence level, and E is the margin of error.

Step 2: Identify the given values. The confidence level is 90%, so the critical value Z can be found using a Z-table or standard normal distribution. For a 90% confidence level, Z ≈ 1.645. The margin of error E is given as 0.03 (3%).

Step 3: Substitute the values into the formula. Use Z = 1.645, E = 0.03, and the maximum variability for the population proportion (p = 0.5, q = 1 - p = 0.5, so p * q = 0.25). The formula becomes n = (1.645^2 * 0.25) / 0.03^2.

Step 4: Simplify the numerator by squaring the Z-value and multiplying it by 0.25. Then, simplify the denominator by squaring the margin of error (E).

Step 5: Divide the simplified numerator by the simplified denominator to calculate the minimum sample size. Always 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

The population proportion refers to the fraction of a population that exhibits a certain characteristic, in this case, U.S. families who eat fast food at least once per week. It is denoted by 'p' and is crucial for estimating how widespread a behavior or opinion is within a defined group.

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to achieve a desired level of precision in estimates. In this scenario, it involves using the desired confidence level (90%) and margin of error (3%) to ensure that the sample accurately reflects the population proportion.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. For this question, a 90% confidence interval means that if the same sampling method were repeated multiple times, 90% of the intervals would contain the true population proportion.

Fast Food You wish to estimate, with 90% confidence, the population proportion of U.S. families who eat fast food at least once per week. Your estimate must be accurate within 3% of the population proportion.a. No preliminary estimate is available. Find the minimum sample size needed.