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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?

Step by step solution

01

Setup hypotheses for marriage support

Understand the hypotheses. Null hypothesis (H0): The proportion of young adults who think their parents would provide marriage support and the proportion of parents who say they would provide marriage support are equal. This can be written as \(p1 = p2\). Alternative hypothesis (H_A): The two proportions are not equal, written as \(p1 ≠ p2\).

02

Calculate test statistic for marriage support

Compute the pooled proportion which is the total number of successful responses divided by the total number of responses, and then calculate the test statistic (Z). For this study: The pooled proportion \(p = (0.41*600 + 0.43*300) / (600 + 300)\) The test statistic is calculated by \[ Z = \frac{p1 - p2}{\sqrt{p * (1- p) * [ 1/n1 + 1/n2 ]}} = \frac{0.41 - 0.43}{\sqrt{p * (1- p) * [ 1/600 + 1/300 ]}} \]

03

Conduct hypothesis test for marriage support

Get the p-value associated with the test statistic from a standard normal (Z) distribution table or use a statistical software to do so. If the p-value is less than or equal to the significance level (usually 0.05), reject the null hypothesis.

04

Setup hypotheses for house buying support

Set up the hypotheses. Null hypothesis (H0): The proportion of parents who would help their children to buy a house or an apartment are equal to or greater than the proportion of young adults who think their parents would provide such support. This can be written as \(p2 ≥ p1\). Alternative hypothesis (H_A): The proportion of parents who would help their children to buy a house or an apartment is less than the proportion of young adults who think their parents would provide such support, written as \(p2 < p1\).

05

Calculate test statistic for house buying support

Same step like the previous test statistic calculation but this time with data on house buying support: \[ Z = \frac{p1 - p2}{\sqrt{p * (1- p) * [ 1/n1 + 1/n2 ]}} = \frac{0.37 - 0.27}{\sqrt{p * (1- p) * [ 1/600 + 1/300 ]}} \]

06

Conduct hypothesis test for house buying support

Get the p-value associated with the test statistic. Since it is a one-tailed test, if the p-value is less than or equal to the significance level (usually 0.05), reject the null hypothesis.

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Key Concepts

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

Independent Random Samples

In statistics, independent random samples are crucial for conducting unbiased hypothesis tests. They refer to two or more samples that are selected from their respective populations in such a way that the selection of a data point in one sample has no influence on the selection of data points in another sample. Moreover, within each sample, the individual observations must be randomly selected and each member of the population must have an equal chance of being included. This ensures that our sample represents the population well and that the comparative analysis between the groups, as in the case of the survey on young adults and parents, is valid and free of confounding variables.

Null Hypothesis

The null hypothesis is a fundamental concept in hypothesis testing, serving as the default or 'status quo' assumption. It asserts that there is no effect or no difference between populations or that any observed effect is due to chance. For instance, in our study concerning financial support for marriage, the null hypothesis posits that the proportion of young adults who believe their parents would provide financial support is the same as the proportion of parents who say they would provide support. In formal terms, we state it as H₀: p1 = p2. The null hypothesis is what we attempt to reject based on our sample data.

Alternative Hypothesis

Contrasting with the null hypothesis is the alternative hypothesis, which suggests there is a statistically significant effect or difference between the populations. It represents the outcome that the study aims to support. From the aforementioned survey, the alternative hypothesis for financial support for marriage proposes that the proportions of young adults and parents differ in their readiness to provide support. Mathematically, this is expressed as H_A: p1 ≠ p2. If we achieve sufficient evidence against the null, we accept the alternative hypothesis.

Pooled Proportion

When comparing two proportions from independent samples, the pooled proportion represents the combined rate of success across both samples. It is calculated by adding the number of successes in both groups and dividing by the total number of observations in both groups. The pooled proportion is used to calculate the standard error of the difference between two sample proportions under the assumption that the null hypothesis is true, as it provides a common estimate of the proportion that applies to both populations. For example, our study's pooled proportion for the belief in financial support for marriage is derived from both the young adults' and parents' responses.

Test Statistics

Test statistics are the standardized values used to determine whether to reject the null hypothesis within the context of a significance level. In the case of comparing two proportions, we often use the Z-test statistic. It quantifies the degree to which the observed data differ from what the null hypothesis would expect. This is calculated by subtracting the difference between the sample proportions and dividing by the standard error. The resulting Z-value allows us to compare our result against a standard normal distribution to find the p-value, which indicates the probability of obtaining a test statistic at least as extreme as ours if the null hypothesis were true.