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The global recession has led more and more people to move in with relatives, which has resulted in a large number of multigenerational households. An October 2011 Pew Research Center poll showed that \(11.5 \%\) of people living in multigenerational households were living below the poverty level, and 14.6\% of people living in other types of households were living below the poverty level (www. pewsocialtrends.org/201 1/10/03/fighting-poverty-in-a-bad- cconomy-americans-move-in-with-relatives/? sre-pre-headline). Suppose that these results were based on samples of 1000 people living in multigenerational households and 2000 people living in other types of households. a. Let \(p_{1}\) be the proportion of all people in multigenerational households who live below the poverty level and \(p_{2}\) be the proportion of all people in other types of households who live below the poverty level. Construct a 98\% confidence interval for \(p_{1}-p_{2}\) - b. Using a \(2.5 \%\) significance level, can you conclude that \(p_{1}\) is less than \(p_{2}\) ? Use the critical-value approach. c. Repeat part b using the \(p\) -value approach.

Step by step solution

01

Calculate sample proportions and standard errors

First, find the sample proportions using the given percentages for multigenerational (\(p_1\)) and other households (\(p_2\)). Next, calculate the standard errors for both the sample proportions.

02

Construct the confidence interval

The confidence interval for the difference in proportions \(p_1 - p_2\) is given by \([p_1 - p_2 \pm z*\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}} ]\) where \(z\) is the z-value from the standard normal distribution for the desired level of confidence, 98% in this case, \(n_1\) is the sample size for multigenerational households, and \(n_2\) is the sample size for other types of households.

03

Perform the Hypothesis Test (Critical Value Approach)

Perform a one-tailed hypothesis test using the critical value approach. The null hypothesis is \(H_0: p_1 = p_2\) and the alternative hypothesis is \(H_1: p_1 < p_2\). Calculate the test statistic for the difference between the two samples. Then compare the test statistic to the critical value from the z-distribution with a significance level of 2.5%. If the test statistic is less than the critical value, reject the null hypothesis.

04

Perform the Hypothesis Test (P-value Approach)

Perform the one-tailed test using the p-value approach. Calculate the test statistic for the difference between the two samples as in Step 3. Then find the p-value, which is the probability of observing a test statistic as extreme as or more extreme than the one calculated, assuming the null hypothesis is true. If the p-value is less than the significance level, 2.5% in this case, reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a method used to determine if there is enough statistical evidence in a sample to conclude something about a population parameter. In our exercise, we're interested in whether the proportion of people below the poverty level in multigenerational households (\(p_1\)) is less than in other households (\(p_2\)).
We start with two hypotheses:

• Null Hypothesis (\(H_0\)): \(p_1 = p_2\), meaning there is no difference in poverty levels between the two groups.
• Alternative Hypothesis (\(H_1\)): \(p_1 < p_2\), meaning fewer people in multigenerational households live in poverty compared to other households.

This test involves calculating a test statistic, which is then compared to a critical value or used to find a p-value. These values help us decide whether to reject or not reject the null hypothesis.

Confidence Interval

A confidence interval gives us a range of plausible values for a population parameter. In this context, it helps estimate the range for the difference in proportions of poverty levels between the two household types.
To construct a 98% confidence interval for \(p_1 - p_2\), we use the formula:
\[\left( p_1 - p_2 \right) \pm z*\sqrt{\frac{p_1(1-p_1)}{n_1} + \frac{p_2(1-p_2)}{n_2}}\]Here, \(z\) is the z-score corresponding to the 98% confidence level, and \(n_1\), \(n_2\) are sample sizes.
This interval helps us understand if the difference in proportions is statistically significant and not due to random sampling error. If the interval does not contain zero, it suggests a significant difference in proportions.

Proportion Difference

This term refers to the difference between the proportions of two groups. In our example, it is the difference between the proportions of households below the poverty level: \(p_1\) and \(p_2\).
Calculating this involves finding the proportion of individuals below the poverty level in multigenerational households (\(p_1\)) and in other types (\(p_2\)). The formula used is:
\[\text{Difference} = p_1 - p_2\]This calculation is crucial in hypothesis testing and confidence interval estimation because it tells us whether the observed difference is due to chance or if it reflects a true population difference.

Significance Level

The significance level, often denoted as \(\alpha\), is the probability of rejecting the null hypothesis when it is true. It sets the criterion for making a decision in hypothesis testing.
In this exercise, a significance level of 2.5% implies that we have a 2.5% risk of concluding that there is a difference in poverty levels when there is not.
A lower significance level means stricter criteria, often leading to more conservative conclusions. Choosing a significance level depends on the context of the study and the consequences of making a Type I error (rejecting a true null hypothesis). For our test, comparing the p-value to the significance level helps decide whether the observed effect is statistically significant.