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A sample proportion represents a way to understand the fraction of the population that exhibits a particular characteristic, based on a sample. In our context, it's about voters who chose the Democratic candidate. To calculate the sample proportion, you divide the number of favorable outcomes (in this exercise, 660 Democratic votes) by the total number of outcomes (1400 voters total). This gives us \[ p = \frac{660}{1400} \approx 0.4714 \]Thus, about 47.14% of the sampled voters, based on our exit poll, favored the Democratic candidate.
This calculation is essential because it is our best estimate of what we expect to see in the entire population. The sample proportion provides a bridge between our specific data and the larger question about the entire group of voters. It gives a rough estimate but by itself doesn't tell us everything; we need more statistical tools, like the standard error and confidence intervals, to get a fuller picture.

Standard error (SE) is crucial in statistics as it measures the variability or dispersion of the sample proportion obtained from different possible samples. It tells us how much the sample proportion is expected to fluctuate from the actual population proportion. In simple terms, it helps us understand how 'off' our sample statistic might be from reality.
The standard error is calculated with the formula:\[ SE = \sqrt{\frac{p(1-p)}{n}} \]where:\

• \(p\) is the sample proportion (0.4714 in our case),
• \(1-p\) is the proportion for all other possibilities (0.5286),
• \(n\) is the sample size (1400)

Plugging in the values gives us \[ SE = \sqrt{\frac{0.4714 \times 0.5286}{1400}} \approx 0.0133 \]This means that if we took multiple samples, we'd expect the sample proportion to vary by about 1.33% from the true population proportion. SE is a key component in constructing confidence intervals, helping us to judge how reliable our sample estimates are.

The z-score is a statistical measure that represents the number of standard deviations a data point is from the mean. It is critical when calculating confidence intervals as it scales the standard error to reflect the desired level of confidence. Different confidence levels have corresponding z-scores that expand or narrow the confidence interval.

• For a 95% confidence interval, the z-score is approximately 1.96.
• For a 99% confidence interval, the z-score is approximately 2.576.

To apply the z-score, we multiply it by the standard error to determine by how much we should adjust our sample proportion to determine the upper and lower bounds of our confidence interval.
For a 95% confidence interval, we use:\[ 0.4714 \pm (1.96 \times 0.0133) = (0.4453, 0.4975) \]For a 99% confidence interval:\[ 0.4714 \pm (2.576 \times 0.0133) = (0.4375, 0.5053) \]These intervals help us predict the likelihood of our sample proportion truly reflecting the population proportion. A wider interval, as seen with the 99% confidence interval, means a greater need for evidence before confidently making any predictions about the winner of the election.