91Ó°ÊÓ

The first step in determining a confidence interval is to calculate the proportion of interest from the sample data. In the given problem, we are interested in the proportion of potential jurors who are regular watchers of crime-scene investigation TV series. Let's break down the process:

• Numerator: The number of successful observations, in this case, jurors who regularly watch, is 350.
• Denominator: The total sample size from which the successful observations are drawn, which is 500.

To get the proportion (\( p \)), simply divide the number of watchers by the total sample size: \( p = \frac{350}{500} = 0.7 \).
This result means that 70% of the surveyed potential jurors regularly watch at least one crime-scene forensics series. Understanding the proportion is crucial, as it forms the foundation for further calculations, like the standard error and the confidence interval.

The standard error is a key statistic used to measure the variability of a sample proportion. It helps us understand how much the sample proportion (\( p \)) might vary from the true population proportion. The formula for standard error when dealing with proportions is: \( SE = \sqrt{\frac{p(1-p)}{n}} \), where \( n \) is the total number in the sample.
For the juror problem, plug in the numbers:\[ SE = \sqrt{\frac{0.7(1-0.7)}{500}} = \sqrt{\frac{0.21}{500}} = 0.02236 \]
This standard error of approximately 0.02236 tells us that the proportion we calculated from the sample (0.7) might vary by this amount if we took other, similar samples from the same population.
In simpler terms, the smaller the standard error, the more precise our estimate. Small standard errors imply that the sample proportion is a good estimate of the true population proportion.

The Z-score is an important statistical tool that helps us determine the confidence interval for our sample proportion. It indicates how many standard deviations an element is from the mean. In the context of confidence intervals, the Z-score corresponds to the desired level of confidence.

• For a 95% confidence interval, the Z-score is approximately 1.96.
• This means we are using 1.96 standard deviations around our sample proportion to create our confidence interval.

The formula for the confidence interval is: \( p \pm Z \times SE \). Plugging in the numbers:\[ 0.7 \pm 1.96 \times 0.02236 = (0.6568, 0.7432) \]
This calculation gives the range within which we are 95% confident that the true proportion of all potential jurors lies. Choosing a 99% confidence interval instead would require a larger Z-score (approximately 2.576), resulting in a wider interval. This wider range reflects the greater certainty we want, allowing more room for the true proportion.