"[NW] Determining Sample Size A physical therapist wants to determine the difference in the proportion of men and women who participate in regular, sustained physical activity. What sample size should be obtained if she wishes the estimate to be within 3 percentage points with 95% confidence, assuming that
b. she does not use any prior estimates?"

Verified step by step guidance

1. Identify the confidence level and margin of error. Here, the confidence level is 95%, and the margin of error (E) is 3 percentage points, which should be expressed as a decimal: \(E = 0.03\).
2. Since no prior estimates of the population proportions are available, use the most conservative estimate for the proportion, which is \(p = 0.5\). This maximizes the product \(p(1-p)\) and thus the required sample size.
3. Find the critical value \(z\) corresponding to the 95% confidence level. This value comes from the standard normal distribution and is typically \(z = 1.96\) for 95% confidence.
4. Use the formula for the sample size needed to estimate a population proportion within a specified margin of error: \( n = \left( \frac{z}{E} \right)^2 \times p(1-p) \)
5. Substitute the values \(z = 1.96\), \(E = 0.03\), and \(p = 0.5\) into the formula to calculate the required sample size \(n\). Remember to round up to the next whole number since sample size must be an integer.

Key Concepts

Sample Size Determination for Proportions

Sample size determination involves calculating the number of observations needed to estimate a population parameter with a desired level of precision and confidence. For proportions, the sample size depends on the margin of error, confidence level, and variability in the population. Larger sample sizes reduce the margin of error, improving estimate accuracy.

Margin of Error and Confidence Interval

The margin of error defines the range within which the true population parameter is expected to lie with a certain confidence level, such as 95%. It reflects the maximum allowable difference between the sample estimate and the true proportion. Confidence intervals provide a range of plausible values for the parameter based on the sample data.

Using Conservative Estimates Without Prior Information

When no prior estimate of the population proportion is available, a conservative approach assumes the maximum variability at p = 0.5. This maximizes the product p(1-p), leading to the largest required sample size to ensure the desired precision. This approach safeguards against underestimating the needed sample size.