Question

The formula used to calculate a large-sample confidence interval for \(p\) is $$ \hat{p} \pm(z \text { critial value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ What is the appropriate \(z\) critical value for each of the following confidence levels? a. \(95 \%\) b. \(98 \%\) c. \(85 \%\)

Short answer

The z critical values correlate with the confidence levels as follows: For a 95% confidence level, the Z value is 1.96. For a 98% confidence level, it is 2.33. And for an 85% confidence level, it is 1.44.

Step-by-step solution

Understand Critical Value Z

The critical value or Z score corresponds to the confidence level for a parameter, which is often found using a z-table. The level of confidence corresponds to the probability that the confidence interval contains the population parameter. This results in a two-sided confidence level since this probability is split evenly on both sides of the mean.

Determine the Z Critical Value for 95% Confidence Level

For a 95% confidence interval, the z critical value is approximately 1.96. This means that if the real parameter of the population is within 1.96 standard deviations from the sample parameter, then the population parameter is within the 95% confidence interval.

Determine the Z Critical Value for 98% Confidence Level

For a 98% confidence level, the critical z value is approximately 2.33. This implies that if the population parameter is within 2.33 standard deviations from the sample parameter, this population parameter falls within the 98% confidence interval.

Determine the Z Critical Value for 85% Confidence Level

For an 85% confidence level, the critical z value is approximately 1.44. This means that if the population parameter is within 1.44 standard deviations of the sample parameter, the population parameter lies within the 85% confidence interval.