7.1.4 Estimating a Population Proportion - Margin of Error and Computing Confidence Intervals

This paragraph introduces the concept of the margin of error and confidence intervals for estimating a population proportion, denoted as 'p'. It explains that the sample proportion, 'p-hat', is an unbiased estimator of 'p', meaning its expected value is equal to the population proportion. The paragraph also discusses the normal distribution of the sampling distribution of 'p-hat' and how it can be converted to a standard normal distribution for easier analysis. The margin of error is introduced as the difference between the mean of the sampling distribution and the upper bound indicated by a critical value, which is part of constructing a confidence interval.

The second paragraph delves deeper into the concept of confidence intervals, explaining how they provide an estimated range for the true population proportion 'p'. It discusses the relationship between the sample proportion 'p-hat', the margin of error 'e', and the confidence level (1 - alpha). The paragraph illustrates how to rearrange the inequality to express 'p' in terms of 'p-hat' and the margin of error, thus forming the confidence interval. It emphasizes that the margin of error represents the maximum likely amount of error in the estimate and is calculated using a specific formula involving the critical z-value and the standard deviation of 'p-hat'.

This paragraph focuses on the formula for calculating the margin of error, which is essential for determining the confidence interval. It explains that the margin of error is the maximum likely amount of error between the sample proportion 'p-hat' and the true population proportion 'p'. The formula involves the critical z-value for the given confidence level, the standard deviation of 'p-hat', and the sample size 'n'. The paragraph also discusses how the margin of error changes with different confidence levels and sample sizes, noting that a larger sample size typically results in a smaller margin of error.

The fourth paragraph explores the relationship between sample size, confidence level, and the margin of error. It explains that as the sample size increases, the margin of error decreases, which makes intuitive sense because a larger sample is more representative of the population. Conversely, as the confidence level increases, the margin of error increases because a higher level of certainty requires a wider interval to capture the true population proportion. The paragraph reinforces the importance of understanding these relationships when constructing and interpreting confidence intervals.

The final paragraph outlines the steps to compute and interpret confidence intervals for a population proportion. It emphasizes the importance of checking the requirements for a valid analysis, such as having a simple random sample and meeting the conditions for a binomial distribution. The paragraph then details the process of identifying known values like 'p-hat', 'q-hat', the confidence level, and sample size 'n', followed by computing the critical value and the margin of error. Finally, it describes how to calculate the upper and lower limits of the confidence interval and presents different ways to express the interval, including a range, point estimate with error, and interval notation.

This paragraph presents a practical example of applying the concepts of confidence intervals to real-world data. It uses the data from a study on the accuracy of fast food drive-through orders at McDonald's, where 33 out of 362 orders were inaccurate. The paragraph guides through the process of constructing a 95% confidence interval for the proportion of inaccurate orders, demonstrating how to put the theoretical concepts into practice.