Question

In Exercises 6.34 to 6.36, we examine the effect of different inputs on determining the sample size needed to obtain a specific margin of error when finding a confidence interval for a proportion. Find the sample size needed to give, with \(95 \%\) confidence, a margin of error within \(\pm 6 \%\) when estimating a proportion. Within \(\pm 4 \%\). Within \(\pm 1 \%\). (Assume no prior knowledge about the population proportion \(p\).) Comment on the relationship between the sample size and the desired margin of error.

Short answer

The required sample sizes are respectively 267 for a 6% margin of error, 600 for a 4% margin of error and 9604 for a 1% margin of error. As the margin of error decreases, the necessary sample size increases.

Step-by-step solution

Finding sample size with margin of error 6%

First, let's find the sample size needed for a margin of error of 6% or 0.06. We substitute \( p = q = 0.5 \), \( Z = 1.96 \) and \( E = 0.06 \) into the formula. The calculation is \( n = (1.96^2 * 0.5 * 0.5) / 0.06^2 = 267 \).

Finding sample size with margin of error 4%

Now we find the sample size for a margin of error of 4% or 0.04. We substitute \( p = q = 0.5 \), \( Z = 1.96 \) and \( E = 0.04 \) into the formula. The calculation is \( n = (1.96^2 * 0.5 * 0.5) / 0.04^2 = 600 \).

Finding sample size with margin of error 1%

Finally we determine the sample size for a 1% margin of error. Plugging in for \( p = q = 0.5 \), \( Z = 1.96 \) and \( E = 0.01 \) into the formula, gives \( n = (1.96^2 * 0.5 * 0.5) / 0.01^2 = 9604 \).

Commenting on the relationship between sample size and margin of error

From these calculations, it is clear that as the margin of error decreases (gets more specific), the necessary sample size increases substantially. This is because to get a more precise estimate, we would need more data.