Question

Use StatKey or other technology to generate a bootstrap distribution of sample proportions and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample proportion as an estimate of the population proportion \(p\). Proportion of survey respondents who say exercise is important, with \(n=1000\) and \(\hat{p}=0.753\)

Short answer

The exercise does not provide specific numeric outcomes as it requires statistical software to compute the bootstrap standard error. However, the process involves creating bootstrap samples, calculating their sample proportions, calculating the standard error of these proportions, and comparing this with the standard error calculated using the Central Limit Theorem.

Step-by-step solution

Generate Bootstrap Sample

Use a statistical software or technology tool to generate a numerous bootstrap samples. A bootstrap sample is a sample taken with replacement from the original sample, of the same size as the original sample.

Calculate Bootstrap Proportions

Calculate the bootstrap proportions. In this case, the proportion of respondents who say exercise is important. Repeat this process for all bootstrap samples.

Determine Bootstrap Standard Error

The bootstrap standard error is calculated as the standard deviation of these bootstrap proportions.

Calculate Standard Error using CLT

The standard error using the CLT can be calculated by the formula \(SE=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\), where \(\hat{p}\) is the sample proportion (0.753 in this case) and \(n\) is the sample size (1000 in this case).

Compare Bootstrap Standard Error and CLT Standard Error

Once both standard errors, from bootstrap and from CLT, are calculated, they can be compared. The value should be relatively close, due to the theoretical base provided by the CLT.