Question

Reported that the percentage of U.S. residents living in poverty was \(12.5 \%\) for men and \(15.1 \%\) for women. These percentages were estimates based on data from large representative samples of men and women. Suppose that the sample sizes were 1,200 for men and 1,000 for women. You would like to use the survey data to estimate the difference in the proportion living in poverty for men and women. (Hint: See Example 11.2) a. Answer the four key questions (QSTN) for this problem. What method would you consider based on the answers to these questions? b. Use the five-step process for estimation problems \(\left(\mathrm{EMC}^{3}\right)\) to calculate and interpret a \(90 \%\) large- sample confidence interval for the difference in the proportion living in poverty for men and women.

Short answer

The estimated difference in poverty proportions between men and women ranges from -0.004 to 0.056, according to a 90% confidence interval.

Step-by-step solution

- Answer the QSTN questions

The QSTN questions in statistics are: Quantity? Source of data? Target population? Nature of result? The Quantity is the difference in proportions of poverty between men and women. The Source of data comes from large representative samples of men and women. The Target population is all U.S. residents. The Nature of result is an estimation problem where you are asked to estimate a numerical parameter.

- Identify method

Based on these answers, use a method to compare two independent proportions, since the estimates are based on two separate samples of men and women.

- Use EMC^3 for estimation problems

The EMC^3 formula stands for Estimate, Margin (of error), Confidence (level), Cutoff, Criteria. The Estimate is the difference in sample proportions i.e. \(0.151 - 0.125 = 0.026\). The Margin of error can be calculated using the formula for standard error for difference between proportions: \(\sqrt{(p1 (1-p1) / n1) + (p2 (1-p2) / n2)} = \sqrt{(0.151 × 0.849 / 1000) + (0.125 × 0.875 / 1200)} ≈ 0.018\). For the Confidence level of 90%, use the critical z value of 1.645. The Cutoff is calculated by multiplying the critical z value with the Margin of error: \(1.645 × 0.018 ≈ 0.030\). For Criteria, refer to standard conditions for inference for difference in proportions. Check if the samples are random, if they are large enough, and if the populations are at least 10 times as large as the samples.

- Calculate and interpret confidence interval

The 90% confidence interval for the difference in proportions is: \(Estimate ± Cutoff = 0.026 ± 0.030\). This interval ranges from -0.004 to 0.056. This means that we are 90% confident that the true difference in the proportion of poverty between men and women in the U.S. falls between -0.004 and 0.056.