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The sample proportion is a way to express how many respondents in a sample group display a certain attribute. It helps us understand the pattern within a specific subset of a larger population. In the context of this exercise, we are looking at judges who favor the death penalty based on their responses.
To calculate the sample proportion, count the number of 'Yes' responses, which indicates favorability of the death penalty, and divide this by the total number of responses. Suppose out of the 15 judges surveyed, 8 said 'Yes'. The sample proportion (denoted as \(\hat{p}\)) is computed as:

• \(\hat{p} = \frac{8}{15}\)

This figure will give us an insight into the approximate proportion of the entire population (all judges) that might share this view. Notice how easily the sample proportion summarizes the data encountered in the survey and acts as a crucial stepping stone towards understanding larger population characteristics.

The population proportion is an estimated or true proportion of a characteristic for the entire population, not just those surveyed. It tells us the percentage of the overall group that shares a specific attribute, like judges who favor the death penalty in this case.
In our given exercise, we wish to estimate the population proportion based on our sample proportion. The calculated sample proportion acts as a point estimate for the population proportion. However, since it's derived from a sample, there's always some level of uncertainty.
We use techniques like confidence intervals, explained below, to provide a sense of how close our sample proportion might be to the true population proportion. It's important to remember that while the sample can guide us, it's not a guaranteed measure of the entire population.

A Z-score is a statistical measure that tells us how many standard deviations a data point is from the mean. It's used here to determine a confidence level for our interval estimate of the population proportion.
When creating a confidence interval, the Z-score helps determine the margin of error, allowing us to account for sampling variability. For a 95% confidence interval, which is a common choice, the Z-score is approximately 1.96.

• This means if we repeated the sampling process many times, about 95% of the calculated intervals would contain the true population proportion.

In simpler terms, the Z-score adds precision to our estimate by quantifying the certainty of our confidence interval. By using the Z-score in the formula \(\hat{p} \pm z\sqrt{\frac{(\hat{p}(1-\hat{p}))}{n}}\), we adjust our interval range, ensuring it accurately reflects the confidence level we've set.

The point estimate is an approximate value that serves as the best single estimate of an unknown population parameter. In our case, the parameter of interest is the population proportion of judges who support the death penalty.

• The sample proportion \(\hat{p}\) is used as a point estimate for the population proportion \(p\).

It's crucial to note that a point estimate is simply an educated guess derived from sample data. While it provides a good approximation, it inherently carries uncertainty because it doesn’t account for sampling variability. However, when coupled with confidence intervals, it gives a fuller picture of our estimate's accuracy. In summary, the point estimate gives us a numerically simple and effective measure of central tendency for the characteristic being studied but must always be interpreted within the context of its associated confidence interval.