How to construct a confidence interval for the proportion, an example

The video script begins by presenting a statistical problem based on a survey of a thousand married men, revealing that 56% have been unfaithful at least once. The task is to form a 95% confidence interval to estimate the true proportion of unfaithful married men. The approach involves a four-step procedure similar to that used for estimating means, but tailored for proportions. Key elements identified include the sample size (N=1000), the sample proportion of unfaithful men (P hat = 0.56), and the complementary proportion (Q hat = 1 - P hat = 0.44). The confidence level is set at 95%, which translates to an alpha value of 0.05, and subsequently, alpha/2 = 0.025. The next step involves obtaining a Z-score from the Z-table for the given alpha/2 at an infinite sample size, which is found to be 1.96. This Z-score is crucial for calculating the margin of error in the subsequent steps.

Continuing from the previous paragraph, the script explains the calculation of the margin of error using the Z-score obtained (1.96), the square root of the product of P hat and Q hat (√(0.56 * 0.44)), and the sample size (N=1000). The margin of error is calculated as 1.96 * √(0.56 * 0.44) / 1000, resulting in a value of approximately 0.03067. This value is then used to determine the confidence interval by adding and subtracting it from the sample proportion (P hat = 0.56). The resulting interval is from 0.53067 to 0.5891, which translates to 52.97% to 59.09% when expressed as percentages. The script concludes by stating that we are 95% confident that the true proportion of married men who are unfaithful lies within this interval. This interval suggests that a majority of married men have been unfaithful at least once, as even the lower bound of the interval is above 50%.