Provude the example of chi test

Answer 1

Sure! Here's an example of how the chi-square test can be used:

Let's say we have a sample of 100 individuals, and we want to determine if there is a relationship between their gender and their preferred mode of transportation (car, bike, or public transportation). We survey the individuals and record their responses.

Here is the data we collect:

Gender: Female (F) or Male (M)
Preferred Mode of Transportation: Car (C), Bike (B), or Public Transportation (PT)

F M
C: 20 25 (total = 45)
B: 15 30 (total = 45)
PT: 10 10 (total = 20)

To perform the chi-square test, we first set up our null hypothesis (H0) and alternative hypothesis (Ha). In this case, our null hypothesis is that gender and preferred mode of transportation are independent, meaning that there is no relationship between them. Our alternative hypothesis is that there is a relationship between gender and preferred mode of transportation.

Next, we calculate the expected frequencies for each cell in the table assuming independence. To do this, we use the formula: Expected frequency = (row total * column total) / sample size.

Expected frequencies:

F M
C: 24.75 20.25
B: 24.75 20.25
PT: 9.5 10.5

Now, we can calculate the chi-square statistic using the formula: chi-square = Σ((observed frequency - expected frequency)^2) / expected frequency.

Calculating chi-square:

chi-square = [(20-24.75)^2/24.75] + [(25-20.25)^2/20.25] + [(15-24.75)^2/24.75] + [(30-20.25)^2/20.25] + [(10-9.5)^2/9.5] + [(10-10.5)^2/10.5]

chi-square = 0.565 + 1.383 + 2.571 + 4.148 + 0.053 + 0.048

chi-square = 8.768

Finally, we compare the calculated chi-square value to the critical chi-square value from the chi-square distribution table at a given significance level (e.g., α = 0.05). If the calculated value exceeds the critical value, we reject the null hypothesis. If the calculated value is less than the critical value, we fail to reject the null hypothesis.

Note: The critical chi-square value is determined by the degrees of freedom, which in this case is (number of rows - 1) * (number of columns - 1) = (2-1) * (3-1) = 1 * 2 = 2.

If the calculated chi-square value of 8.768 exceeds the critical chi-square value at α = 0.05 with 2 degrees of freedom, we can conclude that there is a significant relationship between gender and preferred mode of transportation.