Question

for the following statement, explain the effect on the margin of error and hence the effect on the accuracy of estimating a population mean by a sample mean
decreasing the confidence level while keeping the same sample size
decreasing the confidence level while keeping the same sample size the margin of error and, hence, the accuracy of estimating a population
mean by a sample mean.

Explanation:

The formula for the margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). When the confidence level decreases, the value of \(z_{\alpha/2}\) (the critical value) decreases. Since \(E\) is directly proportional to \(z_{\alpha/2}\) (with \(\sigma\) and \(n\) constant), the margin of error decreases. A smaller margin of error means that the sample mean is likely to be closer to the population mean, so the accuracy of estimating the population mean by the sample mean increases.

Answer:

Decreasing the confidence level while keeping the same sample size decreases the margin of error and, hence, increases the accuracy of estimating a population mean by a sample mean.