The chi-square distribution is a probability distribution that arises when independent standard normal random variables are squared and summed. It is a continuous probability distribution that is widely used in statistical hypothesis testing, particularly in assessing the goodness of fit of observed data to a theoretical distribution, testing the independence of two attributes, and testing the homogeneity of multiple populations.
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The chi-square distribution is a right-skewed distribution, with the degree of skewness decreasing as the number of degrees of freedom increases.
The shape of the chi-square distribution is determined by the number of degrees of freedom, with more degrees of freedom resulting in a distribution that is more symmetric and bell-shaped.
The chi-square test statistic is calculated as the sum of the squared differences between observed and expected values, divided by the expected values.
The chi-square distribution is used to test the goodness of fit of observed data to a theoretical distribution, assess the independence of two categorical variables, and evaluate the homogeneity of multiple populations.
The chi-square test for a single variance is used to determine if the variance of a population is equal to a hypothesized value, and the test statistic follows a chi-square distribution.
Review Questions
Explain how the chi-square distribution is used in the context of contingency tables (3.4 Contingency Tables).
In the context of contingency tables, the chi-square distribution is used to test the independence of two categorical variables. The test statistic is calculated as the sum of the squared differences between the observed and expected cell frequencies, divided by the expected frequencies. The resulting test statistic follows a chi-square distribution with degrees of freedom equal to the number of rows minus 1 multiplied by the number of columns minus 1. If the p-value associated with the test statistic is less than the chosen significance level, the null hypothesis of independence is rejected, indicating that the two variables are related.
Describe the key facts about the chi-square distribution that are relevant to the goodness-of-fit test (11.2 Goodness-of-Fit Test).
The goodness-of-fit test using the chi-square distribution is used to determine whether a set of observed data follows a hypothesized probability distribution. The test statistic is calculated as the sum of the squared differences between the observed and expected frequencies, divided by the expected frequencies. The resulting test statistic follows a chi-square distribution with degrees of freedom equal to the number of categories minus the number of parameters estimated from the data. If the p-value associated with the test statistic is less than the chosen significance level, the null hypothesis that the data follows the hypothesized distribution is rejected, indicating a poor fit.
Analyze how the chi-square distribution is used in the test of independence (11.3 Test of Independence) and the test for homogeneity (11.4 Test for Homogeneity), and explain the key differences between these two tests.
The chi-square test of independence is used to determine whether two categorical variables are independent, while the chi-square test for homogeneity is used to determine whether multiple populations have the same distribution. In the test of independence, the null hypothesis is that the two variables are independent, and the test statistic follows a chi-square distribution with degrees of freedom equal to the number of rows minus 1 multiplied by the number of columns minus 1. In the test for homogeneity, the null hypothesis is that the multiple populations have the same distribution, and the test statistic follows a chi-square distribution with degrees of freedom equal to the number of populations minus 1 multiplied by the number of categories minus 1. The key difference is that the test of independence examines the relationship between two variables, while the test for homogeneity examines the similarity of multiple populations.
The number of independent values or observations that can vary in the final computation of a statistic, which determines the shape and scale of the chi-square distribution.
The statement of no difference or no effect that is tested using the chi-square distribution, and is either rejected or not rejected based on the test results.
The probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true, which is used to determine the statistical significance of the test.