Chi-Square Distribution

Definition

The chi-square distribution is a probability distribution used in statistical hypothesis testing to determine the likelihood of observing a particular set of data given a specific null hypothesis. It is a continuous probability distribution that is derived from the sum of the squares of independent standard normal random variables.

5 Must Know Facts For Your Next Test

1. The chi-square distribution is characterized by a single parameter, the degrees of freedom, which determines the shape of the distribution.
2. The chi-square statistic is calculated by summing the squared differences between observed and expected values, divided by the expected values.
3. The chi-square test is used to assess the goodness-of-fit of a model, test the independence of two categorical variables, and compare observed and expected frequencies.
4. The chi-square distribution is non-negative, right-skewed, and approaches a normal distribution as the degrees of freedom increase.
5. The area under the chi-square curve represents the probability of observing a test statistic at least as extreme as the one calculated from the data, given the null hypothesis is true.

Review Questions

• Explain the purpose of the chi-square distribution in statistical analysis and hypothesis testing.
  • The chi-square distribution is used in statistical hypothesis testing to determine the likelihood of observing a particular set of data given a specific null hypothesis. It is used to assess the goodness-of-fit of a model, test the independence of two categorical variables, and compare observed and expected frequencies. The chi-square statistic is calculated by summing the squared differences between observed and expected values, divided by the expected values. The resulting test statistic is then compared to the chi-square distribution to determine the probability of observing a value at least as extreme as the one calculated from the data, assuming the null hypothesis is true.
• Describe the key features of the chi-square distribution and how they relate to its use in statistical analysis.
  • The chi-square distribution is characterized by a single parameter, the degrees of freedom, which determines the shape of the distribution. The distribution is non-negative and right-skewed, with the shape approaching a normal distribution as the degrees of freedom increase. The area under the chi-square curve represents the probability of observing a test statistic at least as extreme as the one calculated from the data, given the null hypothesis is true. The degrees of freedom are often determined by the number of independent values or observations in the statistical analysis, which is an important factor in interpreting the results of chi-square tests.
• Explain how the chi-square distribution is used in the context of contingency tables and tests of independence.
  • The chi-square distribution is particularly useful in the analysis of contingency tables, which display the frequency distribution of two categorical variables. The chi-square test of independence is used to determine whether two categorical variables are independent or related. The test statistic is calculated by summing the squared differences between the observed and expected frequencies in the contingency table, divided by the expected frequencies. The resulting chi-square statistic is then compared to the chi-square distribution to determine the probability of observing a value at least as extreme as the one calculated, assuming the null hypothesis of independence is true. This allows researchers to draw conclusions about the relationship between the two categorical variables being studied.

Related terms

Goodness-of-Fit Test: A statistical test used to determine whether a set of observed data fits a specified probability distribution.

Contingency Table: A table that displays the frequency distribution of the variables in a sample, often used in chi-square tests of independence.

Degrees of Freedom: The number of independent values or observations that can vary in a statistical analysis, which determines the shape of the chi-square distribution.