Question

put the following in order from narrowest to widest interval. assume the sample size and sample proportion is the same for all four confidence intervals
(a) 93% confidence interval
(b) 83% confidence interval
(c) 89% confidence interval
(d) 99% confidence interval
choose the correct answer below
○ a. (b), (a), (c), (d)
○ b. (d), (c), (a), (b)
○ c. (d), (a), (c), (b)
○ d. (b), (c), (a), (d)

Explanation:

• Confidence interval formula: \(\hat{p}\pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\).
• The width of the confidence interval is \(2z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\).
• For a given \(n\) and \(\hat{p}\), the width depends on \(z_{\alpha/2}\).
• As the confidence level increases, \(z_{\alpha/2}\) increases.
• \(83\%\) confidence level: \(z_{\alpha/2}\) is smaller.
• \(99\%\) confidence level: \(z_{\alpha/2}\) is larger.

Answer:

D. (b), (c), (a), (d)