Refer to the following. In a random sample of 525 teenagers,...

Refer to the following. In a random sample of 525 teenagers, eye colors are cross-classified with favorite colors from among green, red, blue, and purple. Below is a two-way table of the responses. For performing a chi-square test, which of the following is the appropriate null hypothesis? (A) Favorite color choices among all teenagers are $20 \%$ green, $20 \%$ red, $20 \%$ blue, $20 \%$ purple, and $20 \%$ other. (B) There is no difference between the distributions of color choices among teenagers with different eye colors in this sample. (C) There is no difference between the distributions of color choices among teenagers with different eye colors in the population of all teenagers. (D) There is no association between eye color and favorite color among teenagers in this sample. (E) There is no association between eye color and favorite color among teenagers in the population of all teenagers.

Refer to the following. In a random sample of 525 teenagers, eye colors are cross-classified with favorite colors from among green, red, blue, and purple. Below is a two-way table of the responses.
For performing a chi-square test, which of the following is the appropriate null hypothesis?
(A) Favorite color choices among all teenagers are $20 \%$ green, $20 \%$ red, $20 \%$ blue, $20 \%$ purple, and $20 \%$ other.
(B) There is no difference between the distributions of color
choices among teenagers with different eye colors in this sample.
(C) There is no difference between the distributions of color choices among teenagers with different eye colors in the population of all teenagers.
(D) There is no association between eye color and favorite color among teenagers in this sample.
(E) There is no association between eye color and favorite color among teenagers in the population of all teenagers.

Key Concepts

Chi-Square Test for Independence

This is a statistical test used to determine whether there is a significant association between two categorical variables. The test compares the observed counts in each category of a contingency table with the counts expected if the two variables were independent, helping to identify whether any deviations from independence are likely due to chance.

Null Hypothesis in Chi-Square Tests

In the context of a chi-square test, the null hypothesis generally states that there is no association or relationship between the categorical variables in the population, meaning that any observed differences in the sample are due to random variation. It serves as the default assumption that needs to be tested against the alternative hypothesis.

Contingency Tables

Contingency tables, also known as cross-tabulation tables, are used to display the frequency distribution of two categorical variables. They form the basis for the chi-square test for independence by summarizing how the counts of one variable are distributed across the categories of the other variable.

Population vs. Sample Inference

A key concept in statistical testing is distinguishing between conclusions about the sample and inferences about the population. When using chi-square tests, the null hypothesis typically refers to the lack of association in the entire population from which the sample was drawn, rather than just within the sample itself.

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