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 summed. The chi-square distribution is widely used in various statistical analyses, particularly in the context of assessing the goodness-of-fit of observed data to a hypothesized distribution and in the analysis of contingency tables.
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The chi-square distribution is a non-negative continuous probability distribution, meaning its values can never be negative.
The shape of the chi-square distribution depends on the degrees of freedom, which is the number of independent pieces of information or values that can vary in the final computation of the statistic.
As the degrees of freedom increase, the chi-square distribution becomes more symmetric and approaches a normal distribution.
The chi-square statistic is calculated by summing the squared differences between observed and expected values, divided by the expected values.
The chi-square test is used to determine if there is a statistically significant difference between the observed and expected frequencies in one or more categories.
Review Questions
Explain the purpose and use of the chi-square (χ²) distribution in statistical analysis.
The chi-square (χ²) distribution is used in statistical hypothesis testing to assess the goodness-of-fit of observed data to a hypothesized distribution or to analyze the independence of two categorical variables. It is a continuous probability distribution that arises when independent standard normal random variables are squared and summed. The shape of the chi-square distribution is determined by the degrees of freedom, which represent the number of independent pieces of information or values that can vary in the final computation of the statistic. The chi-square test is used to determine if there is a statistically significant difference between the observed and expected frequencies in one or more categories.
Describe the relationship between the chi-square (χ²) distribution and the concept of degrees of freedom.
The degrees of freedom are a key parameter in the chi-square (χ²) distribution, as they determine the shape of the distribution. Degrees of freedom refer to the number of independent pieces of information or values that can vary in the final computation of the statistic. As the degrees of freedom increase, the chi-square distribution becomes more symmetric and approaches a normal distribution. The degrees of freedom are important in determining the critical values used to evaluate the significance of the chi-square test statistic, as the appropriate critical value depends on the specific degrees of freedom associated with the analysis.
Analyze the role of the chi-square (χ²) distribution in the context of the Chi-Square Test of Independence, and explain how it is used to determine the relationship between two categorical variables.
The chi-square (χ²) distribution is central to the Chi-Square Test of Independence, which is used to determine if two categorical variables are independent of each other or if there is a relationship between them. In this test, the chi-square statistic is calculated by summing the squared differences between the observed and expected frequencies in the contingency table, divided by the expected frequencies. The calculated chi-square statistic is then compared to the critical value from the chi-square distribution, with the degrees of freedom determined by the number of rows and columns in the contingency table. If the calculated chi-square statistic exceeds the critical value, the null hypothesis of independence is rejected, indicating a statistically significant relationship between the two categorical variables.
The number of independent values or observations that can vary in the final computation of a statistic. It is a key parameter in the chi-square distribution and determines the shape of the distribution.