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To understand a sample proportion, think of it as a way to represent a part of a whole group. In this context, we are looking at two groups: black women and black men over the age of 65. From these groups, separate samples were taken to determine how many individuals feel vulnerable to crime. - For the women in the sample, 27 out of 56 said they felt vulnerable. The sample proportion is calculated as the number of favorable outcomes divided by the total sample size. So, the proportion of women is: \( \hat{p}_1 = \frac{27}{56} \approx 0.482 \).- Similarly, for the men, 46 out of 63 felt vulnerable, leading to a sample proportion of: \( \hat{p}_2 = \frac{46}{63} \approx 0.730 \).Sample proportions give us a snapshot of what a larger population might be like. However, these are estimates and have a degree of uncertainty, which is where confidence intervals come in to provide more insight.

The standard error is a measure that describes how much variability there is in our sample statistics, like the sample proportion. When dealing with the difference between two sample proportions, the standard error helps us understand how much these sample proportions might vary from the actual population proportions.The formula for the standard error when comparing two sample proportions is:\[ SE = \sqrt{ \frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2} } \]Here:- \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions of women and men, respectively.- \( n_1 \) and \( n_2 \) are their corresponding sample sizes.In our example, plugging in the numbers gives us:\[ SE = \sqrt{ \frac{0.482(1-0.482)}{56} + \frac{0.730(1-0.730)}{63} } \approx 0.0923 \].This standard error indicates the extent of the expected fluctuation between the sample proportions and reflects the precision of our estimate of the difference in proportions.

The difference of proportions provides an easy way to compare two different groups. In the context of feeling vulnerable to crime, we're interested in whether black men or women feel more vulnerable. To find this difference, we subtract one sample proportion from the other:\( \hat{p}_1 - \hat{p}_2 \), which in our case is \( 0.482 - 0.730 = -0.248 \).This result suggests that a lower proportion of women in the sample feel vulnerable compared to the men. The negative sign indicates that the proportion of men who feel vulnerable is higher.Next, we use this difference in combination with the standard error to compute a confidence interval that gives us a range to estimate the true difference in population proportions. A confidence interval helps us see if the observed difference is statistically significant or if it could have occurred by chance.

Statistical significance is a way of determining whether a result from data analysis is likely to be due to something other than random chance. In our problem, we calculated a confidence interval for the difference in proportions that does not include zero:\([-0.3998, -0.0962]\).This interval provides a range of values for the true difference in proportions between black women and men who feel vulnerable.- Since zero is not within our confidence interval, it strongly indicates that there is a statistically significant difference between the two groups.- This finding suggests that our observed difference is unlikely to have happened by random chance. When a result is statistically significant, it helps support a hypothesis or scholarly claim—in this case, suggesting a real difference in the sense of vulnerability towards crime between older black women and men in Atlantic City.