The chi-square (χ²) distribution is a probability distribution used in statistical hypothesis testing. It is a continuous probability distribution that arises when independent standard normal random variables are squared and their sum is taken.
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The chi-square distribution is used to test the null hypothesis that the data follows a specified distribution.
The test statistic, denoted as χ², is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies.
The chi-square distribution has one parameter, the degrees of freedom, which is determined by the number of categories in the data minus 1.
As the degrees of freedom increase, the chi-square distribution approaches a normal distribution.
The chi-square goodness-of-fit test is used to determine if a sample of data is consistent with a hypothesized probability distribution.
Review Questions
Explain the purpose of the chi-square (χ²) distribution and how it is used in statistical hypothesis testing.
The chi-square (χ²) distribution is used in statistical hypothesis testing to determine if a sample of data fits a particular probability distribution. The test statistic, χ², is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies. This test statistic is then compared to a critical value from the chi-square distribution, which is determined by the degrees of freedom. If the calculated χ² value is greater than the critical value, the null hypothesis is rejected, indicating that the sample data does not fit the hypothesized distribution.
Describe the relationship between the chi-square distribution and the degrees of freedom, and explain how this relationship is used in the chi-square goodness-of-fit test.
The chi-square distribution has one parameter, the degrees of freedom, which is determined by the number of categories in the data minus 1. As the degrees of freedom increase, the chi-square distribution approaches a normal distribution. In the chi-square goodness-of-fit test, the degrees of freedom are used to determine the critical value from the chi-square distribution. The test statistic, χ², is then compared to this critical value to determine if the sample data is consistent with the hypothesized probability distribution. The degrees of freedom play a crucial role in this test, as they directly impact the shape and spread of the chi-square distribution, and therefore the decision to reject or fail to reject the null hypothesis.
Analyze the role of the chi-square (χ²) distribution in the context of the chi-square goodness-of-fit test, and explain how the test can be used to evaluate the fit of a sample data to a hypothesized probability distribution.
The chi-square (χ²) distribution is central to the chi-square goodness-of-fit test, which is used to determine if a sample of data is consistent with a hypothesized probability distribution. In this test, the test statistic, χ², is calculated by summing the squared differences between observed and expected frequencies, divided by the expected frequencies. This χ² value is then compared to a critical value from the chi-square distribution, which is determined by the degrees of freedom. If the calculated χ² value is greater than the critical value, the null hypothesis is rejected, indicating that the sample data does not fit the hypothesized distribution. The chi-square distribution, with its unique shape and spread determined by the degrees of freedom, provides the statistical framework for evaluating the fit of the sample data to the hypothesized distribution. By understanding the properties of the chi-square distribution and how it is used in the goodness-of-fit test, researchers can effectively assess the validity of their hypotheses and make informed decisions about the underlying probability distributions of their data.
A method of making decisions using data, where the decisions are based on the likelihood of observing a particular data set under different hypotheses.