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Refer to the following setting. The National Longitudinal Study of Adolescent Health interviewed a random sample of 4877 teens (grades 7 to 12 ). One question asked was "What do you think are the chances you will be married in the next ten years?" Here is a two-way table of the responses by gender: \({ }^{28}\) $$ \begin{array}{lcc} \hline & \text { Female } & \text { Male } \\ \text { Almost no chance } & 119 & 103 \\ \text { Some chance, but probably not } & 150 & 171 \\ \text { A 50-50 chance } & 447 & 512 \\ \text { A good chance } & 735 & 710 \\ \text { Almost certain } & 1174 & 756 \\ \hline \end{array} $$ The appropriate null hypothesis for performing a chi-square test is that (a) equal proportions of female and male teenagers are almost certain they will be married in 10 years. (b) there is no difference between the distributions of female and male teenagers' opinions about marriage in this sample. (c) there is no difference between the distributions of female and male teenagers' opinions about marriage in the population. (d) there is no association between gender and opinion about marriage in the sample. (e) there is no association between gender and opinion about marriage in the population.

Step by step solution

01

Understand the Chi-Square Test Objective

The chi-square test of independence is used to determine if there is an association between two categorical variables. In this case, the variables are gender and opinion about future marriage.

02

Identify the Null and Alternative Hypotheses

The null hypothesis ( H_0 ) generally states that there is no association between the variables being studied. Based on the context of the chi-square test, we can identify the null hypothesis related to gender and marriage opinion in the population.

03

Interpret the Choices

Each choice provides a different interpretation of what the null hypothesis could be. We need to select the one that reflects no association between the variables in the context of the population as the chi-square test typically concerns population-level inferences.

04

Determine the Correct Answer

The correct null hypothesis that matches the scenario of a chi-square test for independence is option (e): "there is no association between gender and opinion about marriage in the population." This states that gender and opinion on marriage are independent in the population, which aligns with the objective of the chi-square test.

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Key Concepts

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

Categorical Variables

Categorical variables are variables that classify data into distinct categories or groups. These are typically qualitative in nature, rather than numerical. For example, in a survey about opinions on marriage, gender is a categorical variable, with categories like 'male' and 'female'. Another categorical variable in this context is the opinion on the likelihood of getting married, with categories like 'almost no chance', 'some chance', 'a 50-50 chance', 'a good chance', and 'almost certain'.

Understanding categorical variables is key because they allow us to segment data in ways that can reveal patterns and relationships between these segments. For instance, by examining responses separately for males and females, researchers can investigate if there's a gender-based difference in marriage expectations.

Null Hypothesis

In statistics, a null hypothesis is a kind of default position that indicates no association between the variables under study. It assumes that any kind of difference or significance seen in the data is due to chance. In the context of chi-square tests for independence, a typical null hypothesis would state that there is no association between the categorical variables being examined.

For instance, when considering the question of whether gender influences opinions about marriage, the null hypothesis would be that gender and opinion are independent, meaning one's gender does not affect or change one's opinion on marriage likelihood. This provides a starting point for statistical analysis, as the chi-square test will determine whether there is enough evidence to reject this assumption.

Statistical Independence

Statistical independence refers to a scenario where the occurrence of one event does not affect the probability of another event happening. In the context of a chi-square test for independence, two categorical variables are said to be statistically independent if the distribution of one variable is the same across the levels of the other variable.

In our example, we're testing whether gender and opinion about future marriage are independent. If they are, knowing someone's gender wouldn’t provide any additional information about their opinion on the likelihood of marriage. The chi-square test helps us assess this by comparing observed frequencies in a contingency table to expected frequencies, assuming independence. If observed and expected frequencies differ significantly, it suggests a possible association.

Population Inference

Population inference refers to the process of using data from a sample to make generalizations about a larger population. In statistical tests like the chi-square test, we often analyze data from a sample and then try to infer if the patterns observed apply to the larger group as a whole.

For example, by using responses from 4877 teens regarding marriage expectations, researchers can infer whether these patterns likely hold true for all teens in the same age range. The null hypothesis often frames these inferences; in this context, it would suggest that any observed association between gender and marriage likelihood in the sample is likely to apply to the larger population as well. The chi-square test helps validate whether we can confidently make such an inference or whether the pattern seen might be the result of sampling variability.