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A point estimate is a single value used to approximate an unknown population parameter. It's essentially a best guess or estimate derived from sample data. In our context, the point estimate is used to gauge the proportion of corporations that prefer hiring nonsmokers over smokers.Given the exercise, the point estimate for the proportion \( p \) of all corporations preferring a nonsmoking candidate is calculated using the formula:

• \( \hat{p} = \frac{x}{n} \)

where \( x \) is the number of corporations that prefer nonsmokers, and \( n \) is the total number of corporations surveyed.By substituting the given numbers from the survey:

• \( \hat{p} = \frac{250}{1000} = 0.25 \).

This means our point estimate indicates that 25% of the corporations prefer nonsmoking candidates. It is important to note that this estimate is based solely on the sample data and serves as a foundational step for further statistical analysis.

The sample proportion is a vital aspect of statistical survey analysis. It refers to the ratio derived from the sample, describing the fraction of the sample exhibiting a particular characteristic, which in this case is the preference for nonsmoking candidates.From the survey details, the sample proportion \( \hat{p} \) is:

• Calculated with \( \hat{p} = \frac{x}{n} \).
• Here, \( x = 250 \) and \( n = 1000 \), so \( \hat{p} = 0.25 \).

This sample proportion value of 0.25 or 25% offers a direct representation of the observed data in the survey. It reflects the likelihood that an individual corporation, within the sample, prefers a nonsmoker.The sample proportion forms a crucial starting point for creating a confidence interval, which further helps to generalize this likelihood to the broader population. This transition from sample insights to population inferences is fundamental in statistics.

The margin of error quantifies the range in which the true population parameter is expected to fall, with a certain level of confidence. It's a reflection of the plus or minus variance from the point estimate due to sampling variability.In this exercise, the margin of error (MOE) for a 95% confidence interval is calculated using:

• \( MOE = Z \times SE \)

The Z-value for a 95% confidence level is 1.96, and from calculations, the SE (standard error) is approximately 0.0137.Thus, the margin of error is:

• \( MOE = 1.96 \times 0.0137 \approx 0.0268 \)

So, the margin of error is approximately 2.68%, which translates to a 95% confidence that the proportion of corporations that would likely prefer nonsmoking candidates lies in the interval created by adding and subtracting the MOE from the point estimate.

Standard error (SE) represents the degree to which a sample statistic tends to vary from the actual population parameter. It provides insight into the reliability of the sample statistic—here, the sample proportion.In the exercise, the standard error of the sample proportion is computed with the formula:

• \( SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \)

Given that \( \hat{p} = 0.25 \) and the sample size \( n = 1000 \):

• \( SE = \sqrt{\frac{0.25 \times 0.75}{1000}} \approx 0.0137 \)

This small standard error indicates a low level of variability in the sample proportion estimate, enhancing our confidence in the results. The smaller the SE, the more precise the point estimate is seen to be, suggesting that the sample of 1000 corporations provides a good reflection of the entire population's sentiment.