A chi-squared (χ²) test is a statistical test used to determine if there is a significant association between two categorical variables. It is often used to compare observed frequencies with expected frequencies and evaluate whether the observed frequencies deviate significantly from what would be expected under a given hypothesis.
Let's consider a simple example to understand chi-square test step by step:
Example: A researcher wants to examine whether there is a relationship between gender (male or female) and favorite ice cream flavor (chocolate, vanilla, or strawberry). The researcher surveyed 100 individuals and recorded their responses.
Step 1: Set up hypotheses:
- Null Hypothesis (H₀): There is no association between gender and favorite ice cream flavor.
- Alternative Hypothesis (H₁): There is an association between gender and favorite ice cream flavor.
Step 2: Set the significance level (α), which determines the probability of rejecting the null hypothesis when it is true. Let's assume α = 0.05.
Step 3: Create a contingency table:
Construct a contingency table to summarize the observed frequencies of each combination of gender and ice cream flavors. It might look like this:
Chocolate Vanilla Strawberry Total
Male 20 30 10 60
Female 40 15 25 80
Total 60 45 35 140
Step 4: Calculate expected frequencies:
Calculate the expected frequencies for each combination under the assumption that the null hypothesis is true (i.e., no association between gender and ice cream flavor). This is done by multiplying the row total by the column total and dividing by the grand total:
Chocolate Vanilla Strawberry Total
Male 25.71 19.29 14.00 60
Female 34.29 25.71 20.00 80
Total 60 45 35 140
Step 5: Calculate the chi-square test statistic:
The chi-square test statistic is calculated as the sum of the squared differences between the observed and expected frequencies divided by the expected frequencies for each cell. The formula is:
χ² = ∑ ((O - E)² / E)
where O is the observed frequency and E is the expected frequency for each cell. In our example, we have three cells for chocolate, vanilla, and strawberry flavors. So the computation will look like this:
χ² = ((20 - 25.71)² / 25.71) + ((30 - 19.29)² / 19.29) + ((10 - 14)² / 14) + ...
Step 6: Determine the degrees of freedom (df):
The degrees of freedom is calculated as (r - 1) * (c - 1), where r is the number of rows and c is the number of columns in the contingency table. In our example, r = 2 (male and female) and c = 3 (chocolate, vanilla, and strawberry), so df = (2 - 1) * (3 - 1) = 2.
Step 7: Look up the critical value:
Using the chi-square distribution table or a statistical software, find the critical value associated with the degrees of freedom and the chosen significance level (α = 0.05). In our example, with df = 2 and α = 0.05, the critical value is approximately 5.99.
Step 8: Compare the calculated chi-square statistic with the critical value:
If the calculated chi-square statistic is greater than the critical value, we reject the null hypothesis and conclude that there is evidence of an association between gender and favorite ice cream flavor. Otherwise, if the calculated chi-square statistic is less than or equal to the critical value, we do not have enough evidence to reject the null hypothesis.
Step 9: Interpret the results:
Based on the calculated chi-square statistic and comparison with the critical value, we draw our conclusion. If we reject the null hypothesis, we can state that there is an association between gender and favorite ice cream flavor. If we fail to reject the null hypothesis, we cannot conclude that there is an association between the two variables.
Remember that this example provided a basic step-by-step explanation of a chi-square test. In practice, it is always recommended to consult a statistics textbook or statistical software for accurate calculations and to understand the limitations and assumptions of the test.
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