Question

which of the following would result in the narrowest confidence interval? a sample size of 100 with 95% confidence a sample size of 30 with 95% confidence. a sample size of 30 with 99% confidence. a sample size of 100 with 99% confidence. check answer

Explanation:

Step1: Recall the formula for confidence interval

The formula for a confidence interval (for a population mean when population standard deviation \(\sigma\) is known) is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). The margin of error \(E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\). A narrower confidence interval corresponds to a smaller margin of error.

Step2: Analyze the effect of sample size \(n\)

Since \(E\propto\frac{1}{\sqrt{n}}\) (when other factors are constant), a larger sample size \(n\) will decrease the margin of error. So, \(n = 100\) is better than \(n=30\) in terms of reducing the margin of error.

Step3: Analyze the effect of confidence level

The confidence level is related to \(z_{\alpha/2}\). For a 95% confidence level, \(z_{\alpha/2}=1.96\), and for a 99% confidence level, \(z_{\alpha/2}=2.576\). Since \(E\propto z_{\alpha/2}\) (when other factors are constant), a lower \(z_{\alpha/2}\) (corresponding to a lower confidence level) will decrease the margin of error.

Answer:

A sample size of 100 with 95% confidence.