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The difference in proportions involves comparing two groups to find how their proportions vary from each other. In this exercise, we compare the proportion of men living in poverty to that of women. We have the percentages: 12.2% for men and 14.8% for women.
This means that we are calculating the difference in proportions as follows:
- For men: 12.2% or 0.122
- For women: 14.8% or 0.148

To find the difference in the proportions living in poverty between men and women in 2015, simply subtract the two values:
\[\Delta p = p_w - p_m = 0.148 - 0.122 = 0.026\]
This indicates that there is a 2.6% higher proportion of women living in poverty compared to men, based on the sample data.

The standard error measures the variability of a sampling distribution. It's like a buffer zone that tells us how "spread out" our sample is around the true population proportion. For each group, the formula for calculating the standard error of a proportion is:
\[SE = \sqrt{\frac{p(1-p)}{n}}\]
Here:

• \(p\) is the proportion of the sample.
• \(n\) represents the sample size.

We need to calculate the standard error for both men and women:

• For men: \[ SE_m = \sqrt{\frac{0.122(1-0.122)}{1200}} = 0.0096 \]
• For women: \[ SE_w = \sqrt{\frac{0.148(1-0.148)}{1000}} = 0.0104 \]

The smaller the standard error, the more reliable our sample proportion is as an estimator of the true population proportion.

The U.S. Census Bureau collects a broad array of data to understand various demographics, including poverty statistics. For this problem, data from the U.S. Census in 2015 showed the poverty rates for men and women. These statistics are typically gathered from a large and representative sample
— in this case, the sample sizes were 1,200 for men and 1,000 for women.

Such data analysis helps us understand disparities and target interventions. Estimating the difference in poverty rates between genders assists policymakers in making informed decisions. Large samples like these help reduce error and ensure accurate reflection of national trends.
Using census data provides a concrete, reliable foundation for statistical analysis, as it is derived from comprehensive, nationwide surveys.

Estimation in statistics involves predicting or inferring a population parameter based on sample data collected. The primary goal of estimation is to obtain useful information about a population from a subset using statistical inference techniques.

In this problem, we are estimating the difference in poverty proportions between men and women. A confidence interval gives us a range of values that is likely to contain the population parameter. For our analysis, we used a 90% confidence interval.
- **Calculate Margin of Error**: Multiply the standard error of the difference by the critical z-value for 90% confidence: \[ ME = 1.645 \times 0.0142 = 0.0234 \]- **Calculate Confidence Interval**: Subtract and add margin of error from the difference: \[ CI = (0.026 - 0.0234, 0.026 + 0.0234) = (0.0026, 0.0494) \]

This means we are 90% confident that the true difference in poverty rates between men and women in 2015 falls between 0.26% and 4.94%. This interval estimation helps provide more context to our statistical findings and lend statistical weight to the observed differences.