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Proportion comparison is a crucial statistical method used to examine if there are significant differences between two groups based on a particular characteristic.
In the context of the exercise, we are interested in determining if there is a difference between the proportions of domiciled men and homeless men who are currently working.
To achieve this, we assess the sample proportions from the given data: 98 out of 260 domiciled men are working, yielding a proportion of approximately 0.377, whereas 34 out of 112 homeless men are working, resulting in a proportion of approximately 0.304.
These individual sample proportions are key indicators that help us to set the foundation for further statistical testing, providing a basis for the next steps in hypothesis verification.

The z-test is a statistical procedure used to determine whether there is a significant difference between the means or proportions of two groups.
In this case, we employ the z-test to compare these proportions and ascertain statistical significance.
Calculating the z-test statistic involves determining the difference between the two sample proportions and dividing this by the standard error of the difference.

• Formula for the test statistic: \[ z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}} \]
• Here, \( \hat{p} \) is the pooled proportion, which combines successes from both groups divided by the total sample size.

The z-test helps to determine if the observed difference between proportions is large enough to reject the null hypothesis, by calculating and comparing this statistic to a critical value from the z-table.

The significance level, often denoted by \( \alpha \), is the threshold for determining whether a result is statistically significant.
It reflects the probability of rejecting the null hypothesis when it is true, thus defining the risk level you are willing to take in making a type I error.

• Common significance levels used are 0.05, 0.01, and 0.10.
• In our exercise, the significance level is set at 0.01, indicating a stricter criterion for ruling out random chance.

This means we want a 99% confidence that our conclusion is correct. The critical value at \( \alpha = 0.01 \) for a one-tailed z-test is approximately 2.33.
If the z-test statistic exceeds this critical value, it suggests sufficient evidence to reject the null hypothesis.

Hypothesis testing begins with clearly defining the null and alternative hypotheses.
These hypotheses provide a structured framework for analysis and decision-making.

• The null hypothesis \( H_0 \) assumes no effect or difference, serving as a default position. In our exercise, it states that the proportion of working domiciled men is equal to the proportion of working homeless men \( (p_1 = p_2) \).
• The alternative hypothesis \( H_a \) suggests that there is an effect or difference, which we are interested in proving. Here, it indicates that the working proportion is greater in domiciled men \( (p_1 > p_2) \).

Careful formulation of these hypotheses is vital, as they guide the statistical analysis and interpretation of results.
Inferences based on the test outcomes are drawn by comparing the test statistic against critical values, leading to either rejecting the null hypothesis or not.