Question

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 \(\pm 3 \%\) when estimating a proportion. First, find the sample size needed if we have no prior knowledge about the population proportion \(p\). Then find the sample size needed if we have reason to believe that \(p \approx 0.7\). Finally, find the sample size needed if we assume \(p \approx 0.9 .\) Comment on the relationship between the sample size and estimates of \(p\).

Short answer

The sample sizes required at 95% confidence level are: for unknown proportion, 1068; for \(p = 0.7\), 753; for \(p = 0.9\), 492. As the value of \(p\) increases, the required sample size decreases.

Step-by-step solution

Understanding the Information Given

We need to find the sample size required to achieve a margin of error of 3% with a confidence level of 95%. Now, for a confidence level of \(95% \), the z-score, \(z\), is \(1.96\). The equation that is used to calculate sample size, \(n\), given a margin of error, \(E\), and proportion, \(p\), is \(n = (z² * p * (1 - p)) / E²\). In our case, we only have the value for \(z\), which is 1.96, and \(E\), which is 0.03.

Calculate Sample Size for Unknown Proportion (p)

If we have no prior knowledge about the population proportion, we assume that \(p = 0.5\) as this will provide the maximum sample size and ensure that our sample is large enough. Plugging these values into the equation: \(n = (1.96² * 0.5 * (1 - 0.5)) / 0.03² = 1067.111\), we always round up to ensure the sample size is large enough, so we get a sample size of 1068.

Calculate Sample Size for p = 0.7

If we have reason to believe that \(p \approx 0.7\), then we will use this proportion in our formula. Thus the calculation becomes \(n = (1.96² * 0.7 * (1 - 0.7)) / 0.03² = 752.952\). Rounding up, we get a sample size of 753.

Calculate Sample Size for p = 0.9

If we assume \(p = 0.9\), replacing these values into our formula outputs \(n = (1.96² * 0.9 * (1 - 0.9)) / 0.03² = 491.053\). Rounding up, we get a sample size of 492.

Relationship between sample size and estimates of p

From the calculations, it is observed that the higher estimates of \(p\), the lesser the required sample size. This can be explained by the fact that when the population proportion is closer to 0 or 1, we have more certainty and therefore we require a smaller sample size to get a precise estimate.