Chi-square distribution

Definition

The chi-square distribution is a statistical distribution that describes the distribution of a sum of the squares of k independent standard normal random variables. It plays a vital role in hypothesis testing, especially for tests related to independence and goodness-of-fit, helping to determine if observed frequencies differ from expected frequencies.

5 Must Know Facts For Your Next Test

1. The chi-square distribution is skewed to the right and approaches a normal distribution as the degrees of freedom increase.
2. In a goodness-of-fit test, the chi-square statistic is calculated by summing the squared difference between observed and expected frequencies divided by the expected frequencies.
3. For tests of independence, the chi-square statistic assesses whether there is a significant association between two categorical variables in a contingency table.
4. The critical values for the chi-square statistic are determined based on the chosen significance level (alpha) and the degrees of freedom.
5. If the calculated chi-square statistic exceeds the critical value from the chi-square distribution table, the null hypothesis is rejected, indicating a significant difference or association.

Review Questions

• How does the chi-square distribution relate to hypothesis testing for independence?
  • The chi-square distribution is used in hypothesis testing to assess whether two categorical variables are independent. By calculating the chi-square statistic from a contingency table, we compare observed frequencies with expected frequencies under the assumption of independence. If the resulting statistic is significantly large, it suggests that the variables are not independent, leading to rejection of the null hypothesis.
• What role does degrees of freedom play in interpreting results from chi-square tests?
  • Degrees of freedom in chi-square tests represent the number of values that are free to vary when estimating statistical parameters. For goodness-of-fit tests, degrees of freedom are calculated as the number of categories minus one. In tests for independence, they are calculated as (rows - 1) * (columns - 1). The degrees of freedom affect the shape of the chi-square distribution and thus influence critical values for significance testing.
• Evaluate how the chi-square goodness-of-fit test can be applied in real-world scenarios, such as marketing research.
  • The chi-square goodness-of-fit test can be applied in marketing research to determine if customer preferences align with company expectations. For example, if a company predicts that customer preferences for a new product will be evenly distributed among various options, they can collect data on actual sales figures. By applying the chi-square test, researchers can assess whether observed sales differ significantly from expected sales distributions. A significant result could prompt further investigation into customer behavior or adjustments to marketing strategies.

Related terms

Degrees of Freedom: A parameter that indicates the number of independent values or quantities that can vary in an analysis without breaking any constraints.

Goodness-of-Fit Test: A statistical test used to determine how well a sample distribution fits a theoretical distribution, often using the chi-square statistic.

Contingency Table: A table used to display the frequency distribution of variables, often used in chi-square tests for independence.