Astrology A sociologist plans to conduct a survey to estimate the percentage of adults who believe in astrology. How many people must be surveyed if we want a confidence level of 99% and a margin of error of four percentage points?


a. Assume that nothing is known about the percentage to be estimated.

Verified step by step guidance

1. Step 1: Start by identifying the formula for determining the required sample size for estimating a population proportion. The formula is: $n\; =\; (Z²\; *\; p\; *\; (1\; -\; p))\; /\; E²$, where 'n' is the sample size, 'Z' is the z-score corresponding to the desired confidence level, 'p' is the estimated proportion, and 'E' is the margin of error.
2. Step 2: For a 99% confidence level, find the z-score. The z-score for a 99% confidence level is approximately 2.576. This value is derived from the standard normal distribution table.
3. Step 3: Since nothing is known about the percentage to be estimated, assume the most conservative estimate for 'p', which is 0.5. This maximizes the product $p\; *\; (1\; -\; p)$, ensuring the largest sample size.
4. Step 4: Substitute the values into the formula. Use $Z\; =\; 2.576$, $p\; =\; 0.5$, and $E\; =\; 0.04$ (since the margin of error is 4%, or 0.04 in decimal form). The formula becomes: $n\; =\; (2.576²\; *\; 0.5\; *\; (1\; -\; 0.5))\; /\; 0.04²$.
5. Step 5: Simplify the expression step by step to calculate the required sample size. First, square the z-score, then calculate $p\; *\; (1\; -\; p)$, and finally divide by the square of the margin of error. Round up the result to the nearest whole number, as the sample size must be an integer.

Key Concepts

Confidence Level

The confidence level represents the degree of certainty that the true population parameter lies within the confidence interval. A 99% confidence level means that if the survey were repeated multiple times, 99% of the calculated intervals would contain the true percentage of adults who believe in astrology. This high level of confidence indicates a strong assurance in the results, but it also requires a larger sample size.

Margin of Error

The margin of error quantifies the range within which the true population parameter is expected to fall, based on the sample results. In this case, a margin of error of four percentage points means that the estimated percentage of adults who believe in astrology could be four points higher or lower than the survey result. A smaller margin of error requires a larger sample size to achieve greater precision in the estimate.

Sample Size Calculation

Sample size calculation is the process of determining the number of respondents needed to achieve a desired level of confidence and margin of error. When estimating proportions, the formula incorporates the confidence level, margin of error, and an estimate of the population proportion. In this scenario, since no prior estimate is available, a conservative approach is often used, typically assuming a proportion of 0.5 to maximize the required sample size.