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A confidence interval gives us a range of values within which we expect the true difference in proportions to lie, with a certain level of confidence. In our exercise, we want to find this interval for the difference between the proportions of men and women living with parents. The formula for the confidence interval is \[ p_{m} - p_{w} \pm z \cdot \sqrt{\frac{p_{m} \cdot (1 - p_{m})}{n_{m}} + \frac{p_{w} \cdot (1 - p_{w})}{n_{w}}} \] where \( p_{m} \) and \( p_{w} \) are sample proportions, and \( n_{m} \) and \( n_{w} \) are sample sizes for men and women, respectively.

• For men, \( p_{m} = 0.24 \) and \( n_{m} = 800 \).
• For women, \( p_{w} = 0.18 \) and \( n_{w} = 850 \).

The critical value \( z \) for a 95% confidence interval is 1.96. Plugging in these values, the confidence interval we obtain is \( [0.02, 0.1] \). This means that the true difference in the proportion of men and women who live with parents is likely between 2% and 10%, with 95% confidence.
This interval helps us understand the range of possible differences in the population.

Hypothesis testing is a method used to decide whether there is enough evidence to reject a claim about a population. Here, we examine whether there's a difference between men and women living with parents. We start by stating

• Null hypothesis (\( H_0 \)): No difference between proportions,
• Alternative hypothesis (\( H_a \)): Proportions are different.

To test these, we calculate a test statistic using the formula: \[ Z = \frac{(p_{m} - p_{w})}{\sqrt{\frac{p(1 - p)}{n_{m}} + \frac{p(1 - p)}{n_{w}}}} \] Where \( p \) is a pooled proportion from both samples. We then compare this \( Z \) to a critical value derived from a standard normal distribution. At a 2% significance level, the critical value is ±2.33. If our test statistic exceeds this critical range, we reject \( H_0 \) and accept \( H_a \).In this scenario, we calculated that the test statistic did reach our critical range, indicating a statistically significant difference in the living situations of men and women at a 2% level.

The p-value approach of hypothesis testing provides another way to determine the significance of observed data. It tells us how probable our observed data would be if the null hypothesis were true. A smaller p-value means stronger evidence against the null hypothesis.
In our exercise, we want to determine if there is a true difference between men's and women's living situations using this method.

• We calculate a test statistic, usually a \( Z \) score, based on our sample data.
• The p-value is the probability that a test statistic as extreme as our observed value could occur, assuming the null hypothesis is true.

If the p-value is less than the significance level, which is 2% in this context, we reject the null hypothesis. Conversely, if it exceeds 2%, we do not reject it.
In the given exercise, it was found that the p-value was less than 2%, indicating strong evidence against \( H_0 \), allowing us to conclude a significant difference exists between male and female living arrangements.

The difference in proportions is a measure used to compare two populations. In this case, it evaluates the difference between the proportions of men and women living with parents. It is calculated as \( p_{m} - p_{w} \).

• For men, \( p_{m} = 0.24 \)
• For women, \( p_{w} = 0.18 \)

This gives a proportion difference of 0.06 or 6%.
This figure shows how much more likely it is for males within the specified group to live with parents compared to females.
Knowing the proportion difference is crucial as it directly informs us about the comparative characteristics of the two groups.
This helps to understand whether policies or interventions may be needed and where should focus be directed, based on the needs identified.

The significance level (\( \alpha \)) is a threshold we set to decide whether we should reject the null hypothesis. It represents the probability of making a Type I error, that is, rejecting \( H_0 \) when it is actually true.

• In the context of our exercise, the significance level is set at 2% (0.02).

This means if there's truly no difference between the groups, there's only a 2% chance our results would falsely suggest a difference. This is a relatively strict criterion and indicates that we require strong evidence before concluding that men and women differ in their living arrangements.
How low the significance level is set depends on the context and how severe the consequences of a Type I error would be. In scientific studies, a more common significance level is 5%, but stricter levels like 2% are used when accuracy is more important.