Question

The formula used to compute a large-sample confidence interval for \(\pi\) is $$ p \pm(z \text { critical value }) \sqrt{\frac{p(1-p)}{n}} $$ What is the appropriate \(z\) critical value for each of the following confidence levels? a. \(95 \%\) d. \(80 \%\) b. \(90 \%\) e. \(85 \%\) c. \(99 \%\)

Short answer

The z critical values for the confidence levels 95%, 90%, 99%, 80%, and 85% are 1.96, 1.645, 2.575, 1.28, and 1.44 respectively.

Step-by-step solution

Understanding the Confidence Interval

The confidence interval is expressed as \( p \pm z \sqrt{\frac{p(1-p)}{n}} \) where \( p \) is the population proportion, \( n \) is the sample size, and \( z \) is the critical value corresponding to the confidence level. The critical value \( z \) is determined by the desired confidence level and can be looked up in a standard normal distribution table.

Calculating the z critical value for a 95% confidence level

The z critical value for a 95% confidence level is 1.96.

Calculating the z critical value for a 90% confidence level

The z critical value for a 90% confidence level is 1.645.

Calculating the z critical value for a 99% confidence level

The z critical value for a 99% confidence level is 2.575.

Calculating the z critical value for an 80% confidence level

The z critical value for an 80% confidence level is 1.28.

Calculating the z critical value for an 85% confidence level

The z critical value for an 85% confidence level is 1.44.