Question

The report "Young People Living on the Edge" (Greenberg Quinlan Rosner Research, 2008 ) summarizes a survey of people in two independent random samples. One sample consisted of 600 young adults (age 19 to 35 ) and the other sample consisted of 300 parents of children age 19 to \(35 .\) The young adults were presented with a variety of situations (such as getting married or buying a house) and were asked if they thought that their parents were likely to provide financial support in that situation. The parents of young adults were presented with the same situations and asked if they would be likely to provide financial support to their child in that situation. a. When asked about getting married, \(41 \%\) of the young adults said they thought parents would provide financial support and \(43 \%\) of the parents said they would provide support. Carry out a hypothesis test to determine if there is convincing evidence that the proportion of young adults who think parents would provide financial support and the proportion of parents who say they would provide support are different. b. The report stated that the proportion of young adults who thought parents would help with buying a house or apartment was .37. For the sample of parents, the proportion who said they would help with buying a house or an apartment was . \(27 .\) Based on these data, can you conclude that the proportion of parents who say they would help with buying a house or an apartment is significantly less than the proportion of young adults who think that their parents would help?

Short answer

a. There isn't sufficient evidence at the 0.05 significance level to say that the proportion of young adults who think parents would provide financial support for getting married and the proportion of parents who say they would provide such support are different.\n b. There's evidence at the 0.05 significance level to conclude that the proportion of parents who say they would help with buying a house or an apartment is significantly less than the proportion of young adults who think their parents would help.

Step-by-step solution

Set up the Hypotheses

For both parts (a) and (b), the first step is to set up the null and the alternative hypotheses which will guide the problem-solving process. For part (a), the null hypothesis (H0) is that there's no difference in the proportions, i.e., the proportion of young adults who think their parents would provide support equals the proportion of parents who would provide support. The alternative hypothesis (HA) is that the proportions are different. Similarly, for (b), the null hypothesis is that the proportion of parents who say they would help is equal to the proportion of young adults who think they would help, while the alternative hypothesis is that the proportion of parents who say they would help is less than the proportion of young adults.

Compute the Test Statistic and the P-value

Next, calculate the test statistic. Using the given sample sizes and proportions, compute the pooled proportion and then the test statistic using the appropriate formula. Once you have the test statistic, you can determine the p-value using a standard normal distribution (Z-distribution) for both parts (a) and (b).

Conclude the Hypothesis Test

Compare the p-values to the significance level \(\alpha = 0.05\). If the p-value is less than \(\alpha\), reject the null hypothesis. If the p-value is greater than or equal to \(\alpha\), do not reject the null hypothesis. From the p-values, draw conclusions about the results in terms of the original questions asked in the exercise.

Key concepts

Null Hypothesis

The null hypothesis is an essential concept in hypothesis testing. It is a statement that assumes no effect or no difference in the population. In the context of the given exercise, the null hypothesis for part (a) posits that there is no difference between the proportion of young adults who think their parents would provide financial support for getting married and the proportion of parents who say they would provide such support. This hypothesis is symbolically represented as \( H_0: p_1 = p_2 \), where \( p_1 \) is the proportion of young adults and \( p_2 \) is the proportion of parents.

For part (b), the null hypothesis suggests that the proportion of parents who say they would help with buying a house equals the proportion of young adults who think they would receive help, represented as \( H_0: p_1 = p_2 \).

When conducting hypothesis tests, the null hypothesis often serves as the baseline assumption. The goal is to determine if there is enough evidence to reject it, suggesting something significant or different is occurring.

Alternative Hypothesis

The alternative hypothesis stands in contrast to the null hypothesis. It proposes that there is a meaningful difference or effect. In hypothesis testing, this hypothesis is what researchers aim to support.

In part (a) of the exercise, the alternative hypothesis (Ha) is that the proportions differ, which is expressed as \( H_A: p_1 eq p_2 \). This suggests that the viewpoint of young adults about parental support and the parents' own beliefs about their support are divergent.

Meanwhile, for part (b), the alternative hypothesis is specifically that the parents' willingness to help with housing is less than what the young adults perceive. Symbolically, this is represented as \( H_A: p_1 > p_2 \).

The alternative hypothesis is crucial because hypothesis tests aim to determine if there is sufficient evidence to favor this hypothesis over the null. It is the statement of change or difference that researchers seek evidence for.

P-value

The p-value in hypothesis testing is a measure of the strength of the evidence against the null hypothesis. It quantifies the probability of observing the test results, or something more extreme, assuming the null hypothesis is true.

Consider the exercise; once we calculate the test statistic, the p-value helps us determine whether the difference in proportions is statistically significant. A small p-value indicates strong evidence against the null hypothesis. Typically, a significance level \( \alpha \) such as 0.05, is used.

- If the p-value is less than \( \alpha \), we reject the null hypothesis.
- If the p-value is greater than or equal to \( \alpha \), we fail to reject it.

In parts (a) and (b) of the example, the calculations of p-values guide us in deciding if the perceived differences in support between young adults and their parents are significant. The smaller this value, the stronger the case for the alternative hypothesis.

Test Statistic

The test statistic is a crucial component of hypothesis testing. It is a standardized value that helps you determine the position of your sample result relative to the null hypothesis. For comparing two proportions, like in the exercise, the test statistic typically follows a Z-distribution.

To find the test statistic, you first need to calculate the pooled proportion. This is done by combining the successes from both samples and dividing by the total number of observations. The test statistic is then calculated using the formula:
\[ Z = \frac{(\hat{p}_1 - \hat{p}_2)}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}} \]
where \( \hat{p}_1 \) and \( \hat{p}_2 \) are the sample proportions, \( \hat{p} \) is the pooled proportion, and \( n_1 \) and \( n_2 \) are the sample sizes.

The calculated Z-value indicates how far, in standard deviations, the observed difference is from the null hypothesis assumption. By comparing this value to a critical value from the Z-distribution, or by calculating the p-value, we decide whether to reject the null hypothesis.