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The sample size is a crucial aspect of statistical studies. It refers to the number of individual data points collected in a study or survey. In our example, the study used 30,000 Americans.A larger sample size can lead to more accurate results. When a sample size is large, it tends to better represent the population under study and reduces the margin of error.
This is because larger samples provide more information and more reliable statistics. In the formula for a confidence interval, the sample size, represented by the symbol \( n \), appears in the denominator of the error term \( \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \).
As \( n \) increases:

• The error margin decreases.
• The confidence interval becomes narrower.
• We gain more precise estimates of the population parameter (in this case, the proportion of smokers).

This is why, even for a 99.9% confidence level, the interval can remain narrow if the sample size is substantial.

Proportion represents a part of the whole. In statistics, the proportion is often denoted as \( \hat{p} \) and is used to infer about the population based on the sample data. In the exercise, the proportion of smokers in the sample was reported as \(20.4\%\).To calculate the proportion from sample data:

• Count the number of successful outcomes (e.g., people who smoke).
• Divide by the total number of observations in your sample.

For instance, if in our study, 6,120 out of 30,000 people smoked, the sample proportion would be \( \frac{6120}{30000} = 0.204 \) or 20.4%.Once we have the sample proportion, it allows estimation of the population proportion. In our context, it helps estimate what proportion of the entire U.S. adult population might be smokers.
Proportion is an essential component when constructing confidence intervals, ensuring that the measure remains relevant and accurate.

The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. In the context of confidence intervals, the Z-score is linked to the confidence level, which indicates how sure you are that the population parameter lies within the interval.For confidence intervals, the Z-score can be found using a Z-table, which shows the score's relationship to the standard normal distribution. The higher the confidence level, the higher the Z-score.
For instance:

• At 95% confidence, the Z-score is approximately 1.96.
• At 99% confidence, the Z-score is about 2.576.
• At 99.9% confidence, like in our exercise, the Z-score is around 3.291.

In our equation for the confidence interval, \( \hat{p} \pm Z \times \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \), the Z-score adjusts the width of the confidence interval. A larger Z-score indicates a broader reach initially, but when paired with a large sample size, the interval remains manageable.Understanding the Z-score is essential as it helps in grasping how confident we can be that the calculated interval truly contains the population parameter.