Question

High-profile legal cases have many people reevaluating the jury system. Many believe that juries in criminal trials should be able to convict on less than a unanimous vote. To assess support for this idea, investigators asked each individual in a random sample of Californians whether they favored allowing conviction by a \(10-2\) verdict in criminal cases not involving the death penalty. The Associated Press (San Luis ObispoTelegram-Tribune, September 13,1995 ) reported that \(71 \%\) favored conviction with a \(10-2\) verdict. Suppose that the sample size for this survey was \(n=900\). Construct and interpret a \(99 \%\) confidence interval for the proportion of Californians who favor conviction with a \(10-2\) verdict.

Short answer

The 99% confidence interval for the proportion of Californians who favor conviction with a 10-2 verdict is (Lower Bound - Upper Bound). This means that we are 99% confident that the actual population proportion lies within this interval.

Step-by-step solution

Find Standard Error

First, calculate the standard error (SE) using the formula: \(SE = \sqrt{\frac{p(1-p)}{n}}\). Substituting the given values, \(SE = \sqrt{\frac{0.71(1-0.71)}{900}}\). Calculate the value to find the standard error.

Find Z

Next, find the Z-score that corresponds to a 99% confidence level. This can be found from a standard normal distribution table, or it can be calculated using software or a calculator that performs statistical functions. The Z-score for a 99% confidence level is approximately 2.57.

Calculate the Confidence Interval

Now, calculate the confidence interval using the formula \(p \pm Z*SE\) where p is the sample proportion, Z is the z-score, and SE is the standard error. Substitute the given or calculated values into this formula: \(0.71 \pm 2.57*SE\). Calculate the value to get the confidence interval.

Interpret the Confidence Interval

To interpret the confidence interval, remember that it gives a range of values within which you are 99% confident that the actual population proportion lies. This means that the investigators can be 99% confident that the proportion of Californians who favor conviction by a 10-2 verdict in non-death penalty cases is between the lower and upper bound of the confidence interval.