Question

Data from a representative sample were used to estimate that \(32 \%\) of all computer users in 2011 had tried to get on a Wi-Fi network that was not their own in order to save money (USA Today, May 16,2011 ). You decide to conduct a survey to estimate this proportion for the current year. What is the required sample size if you want to estimate this proportion with a margin of error of 0.05 ? Calculate the required sample size first using 0.32 as a preliminary estimate of \(p\) and then using the conservative value of \(0.5 .\) How do the two sample sizes compare? What sample size would you recommend for this study?

Short answer

The required sample sizes for the two cases are 866 and 961, respectively. Given the slight difference and for greater assurance in the results of the survey, a sample size of 961 is recommended.

Step-by-step solution

Find Formula for Sample Size

The formula for finding sample size when you want to estimate a proportion with a certain margin of error is given by \(n = p*(1-p)*(z/E)^2\). Here, 'n' is the sample size, 'p' is the preliminary estimate, 'E' is the margin of error, and 'z' is the z-score for the desired level of confidence.

Calculate Sample Size with p=0.32

In this scenario, 'p' is 0.32, 'E' is 0.05, and 'z' is 1.96 (z-score for 95% confidence level, assuming we want to be 95% confident in our estimate). Using these values in the sample size formula, we get \(n = 0.32*(1-0.32)*(1.96/0.05)^2\). Calculating this gives us \(n ≃ 865.2\). Since we can't have a fractional sample size, we round up to the next whole number, resulting in \(n = 866\).

Calculate Sample Size with p=0.5

Now, we calculate the sample size using the conservative value of 'p'=0.5. Thus, our formula becomes \(n = 0.5*(1-0.5)*(1.96/0.05)^2\). Calculating this yields \(n ≃ 960.4\). Again, rounding up to the next whole number, we get \(n = 961\).

Compare Sample Sizes and Give Recommendation

Note that the sample size required when using 'p'=0.5 is larger than when using 'p'=0.32. This is because a larger 'p' value implies greater variability, and hence, a larger sample size is needed for the same margin of error. The larger sample size can be viewed as a more conservative estimate. Considering the minimal difference in sample size and for greater assurance in the results, it is recommended to use the larger sample size of 961.