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Population proportions represent the ratio of individuals in a specific group who possess a certain characteristic. Understanding this concept is critical when comparing two different groups in a sample.
In our exercise, we are comparing two groups: black women and black men over the age of 65 in Atlantic City. Specifically, we want to know what portion of each group "feels vulnerable" to crime.

• The sample proportion for black women, denoted by \( \hat{p}_1 \), is approximately 0.4821. This means about 48.21% of the sampled women felt vulnerable to crime.
• The sample proportion for black men, denoted by \( \hat{p}_2 \), is approximately 0.7302, or about 73.02% of the men felt vulnerable.

By calculating these proportions, we lay the groundwork for further analysis like constructing confidence intervals.

The standard error (SE) measures the variability or dispersion of the sample estimate and helps us understand how much the estimated proportions might differ from the true population proportions. It is crucial in determining how precise our estimates are.
The formula used in our exercise is:

• Standard Error for difference in proportions: \[ SE = \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} \]

Where \( n_1 = 56 \) and \( n_2 = 63 \), are the sample sizes for women and men, respectively.
Using this formula, we calculate the standard error to be approximately 0.0924. This value shows us the expected variability in the difference between the two population proportions if multiple samples were taken.

The critical value is a factor used to calculate the margin of error in a confidence interval. It is derived from the z-distribution, which assumes a normal distribution of the sample data.
For a 90% confidence level, a standard z-table or calculator is used to determine the critical value. In this instance, the critical value \( z^* \) is about 1.645.
The critical value tells us how many standard deviations away from the mean our bounds should be. It adjusts the width of the confidence interval, providing a range in which we can be fairly certain the true difference in population proportions falls. This concept helps to link the statistical theory with real-world data interpretation.

Interpreting results is the final, yet crucial stage of statistical analysis. It involves understanding what the calculated confidence interval actually means in context.
Our confidence interval for the difference in population proportions is \((0.0961, 0.4001)\). This means we are 90% confident that the true difference in the proportion of men and women who "feel vulnerable" to crime lies between 9.61% and 40.01%.

• The fact that this entire interval is above zero suggests that there is a significant difference between the two groups with regard to feeling vulnerable to crime.
• This implies that, not only are men more likely to feel vulnerable, but we also have sufficient statistical evidence to back this claim based on the data.

Interpreting the results helps us decide if the hypothesis is statistically significant and provides insights that can influence further research or policy-making.