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 lie detector trials in which the technology misses a lie, with \(n=48\) and \(\hat{p}=0.354\)

Short answer

The process involves performing a bootstrap simulation for the provided lie detector data, then calculating the standard errors from it and comparing it with the Central Limit Theorem's estimated standard error. The two calculated standard errors should be similar, demonstrating the Central Limit Theorem's utility.

Step-by-step solution

Generate Bootstrap Distribution

Using StatKey or similar statistical software, simulate 10000 bootstrap samples of size \(n = 48\) from a binomial distribution where \(\hat{p} = 0.354\). StatKey generates the bootstrap distribution for you, which are the sample proportions from the bootstrap samples.

Calculate Standard Error of Bootstrap Distribution

After creating the bootstrap distribution, compute the standard error, which is simply the standard deviation of the bootstrap sample proportions. This calculation is also usually completed by the software used.

Calculate Standard Error via Central Limit Theorem

The standard error provided by the Central Limit Theorem can be calculated using the formula: \(SE = \sqrt{\hat{p}(1-\hat{p})/n}\). Where \(\hat{p}\) is the sample proportion and \(n\) is the sample size. Substitute \(\hat{p} = 0.354\) and \(n = 48\) into this formula to get the Central Limit Theorem estimated standard error.

Comparing the Two Standard Errors

For the final step, consider the standard error calculated from the bootstrap distribution and the one calculated by the Central Limit Theorem. Generally, both values should be quite comparable, confirming the efficiency of the Central Limit Theorem as a tool for estimating standard errors.