91Ó°ÊÓ

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: \({ }^{28}\) $$ \begin{array}{lcc} \hline & \text { Female } & \text { Male } \\ \text { Almost no chance } & 119 & 103 \\ \text { Some chance, but probably not } & 150 & 171 \\ \text { A 50-50 chance } & 447 & 512 \\ \text { A good chance } & 735 & 710 \\ \text { Almost certain } & 1174 & 756 \\ \hline \end{array} $$ (a) A Type I error is possible. (b) A Type II error is possible. (c) Both a Type I and a Type II error are possible. (d) There is no chance of making a Type I or Type II error because the \(P\) -value is approximately \(0 .\) (e) There is no chance of making a Type I or Type II error because the calculations are correct. For these data, \(\chi^{2}=69.8\) with a \(P\) -value of approximately \(0 .\) Assuming that the researchers used a significance level of \(0.05,\) which of the following is true?

Step by step solution

01

Understand the hypotheses

In the context of chi-square tests, the null hypothesis (\(H_0\)) often states there is no association between two categorical variables. Here, it means gender and the belief in marriage chances are independent.

02

Define Type I and Type II errors

A Type I error occurs when we reject a true null hypothesis. Conversely, a Type II error happens when we fail to reject a false null hypothesis.

03

Evaluate the P-value

A P-value is used to test the null hypothesis. If the P-value is less than the significance level \( \alpha = 0.05 \), we reject the null hypothesis. Here, the P-value is approximately 0, which is less than 0.05.

04

Conclusion based on P-value

Since the P-value is less than 0.05, we reject the null hypothesis. This means there is evidence to suggest a relationship between gender and belief in marriage chances.

05

Determine possibility of errors

Based on the rejection of the null hypothesis, there is a chance of making a Type I error since we might have rejected a true null hypothesis. However, since we did not fail to reject the null hypothesis, a Type II error is not possible.

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Key Concepts

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

Type I Error

In statistical hypothesis testing, a Type I error occurs when you reject a true null hypothesis. This is often regarded as a false positive. Imagine that the null hypothesis is true, indicating no association between the categorical variables under investigation—in this case, gender and belief in future marital status. - When performing a chi-square test, if the P-value is lower than the chosen significance level (like 0.05), you reject the null hypothesis.
- However, this rejection might be incorrect, thus resulting in a Type I error, where you conclude incorrectly that there is an association when in fact, there isn't. The consequence of a Type I error in social sciences could lead to believing there's a gender influence when none exists, potentially affecting policies or interventions that might have been based on this conclusion.

Type II Error

A Type II error, also known as a false negative, occurs when you fail to reject a false null hypothesis. In this scenario, you miss detecting an actual effect or association between the categorical variables assessed—in this case, gender and marriage belief. - This means accepting the null hypothesis when gender does indeed influence marital belief, but this influence remains undetected.
- A Type II error can occur for various reasons, such as insufficient sample size or variability in the data that isn't captured by the test. In this exercise, a Type II error is not considered possible because the null hypothesis was actually rejected based on a very low P-value. Thus, gender is suggested to influence beliefs about future marriage, and failing to detect this effect wasn't an issue in this case.

P-value

The P-value in statistical testing is a measure that helps you determine the strength of your evidence against the null hypothesis. Consequently, it helps you decide whether to reject or fail to reject the null hypothesis. - If the P-value is less than or equal to your significance level (commonly 0.05), it indicates strong evidence to reject the null hypothesis.
- A small P-value suggests that the observed data is unlikely under the assumption that the null hypothesis is true. In the given exercise, the P-value was approximately 0, much lower than 0.05. This indicates very strong evidence against the null hypothesis, leading to its rejection, and implying a significant relationship between gender and belief in marriage likelihood.

Categorical Variables

Categorical variables are those that represent distinct categories or groups. They are not numerical but rather qualitative in nature, like gender (male or female) in this context. - Categorical variables can be nominal with no order (like color) or ordinal where a clear order exists (like preference levels).
- The chi-square test, used in this scenario, specifically analyzes the relationship between two categorical variables. In the exercise, the two categorical variables were 'gender' and 'belief in marriage chances,' both non-numerical but critically important in examining if there's an association between them. Understanding these variables helps in analyzing survey or observational study data like that from the Longitudinal Study of Adolescent Health, where associations between different demographic factors are explored.