A sample of 500 drivers was asked whether or not they speed while driving. The following tables gives a two-way classification:
Speed Never speed
Male 222 78
Female 128 72
We wish to test whether gender and speeding are related at the 1% significance level.
To test whether gender and speeding are related at the 1% significance level, we can use the chi-square test of independence. This test allows us to determine if there is a relationship between two categorical variables.
To perform the chi-square test of independence, we need to follow these steps:
Step 1: Set up hypotheses
- Null hypothesis (H0): Gender and speeding are independent (There is no relationship between gender and speeding).
- Alternative hypothesis (H1): Gender and speeding are dependent (There is a relationship between gender and speeding).
Step 2: Calculate the expected frequencies
- Calculate the expected frequencies for each cell assuming the null hypothesis is true. The expected frequency for each cell is calculated as (row total * column total) / grand total.
In this case, the row totals and column totals are as follows:
- Row totals: Male - 300, Female - 200
- Column totals: Speed - 350, Never speed - 150
- Grand total: 500
For example, the expected frequency for the cell Male and Speed can be calculated as (300 * 350) / 500 = 210.
Calculate the expected frequencies for all cells:
Speed Never speed Row Total
Male 210 90 300
Female 140 60 200
Column Total 350 150 500
Step 3: Calculate the chi-square statistic
- Compute the chi-square statistic using the formula:
χ2 = Σ [(O - E)² / E], where O is the observed frequency and E is the expected frequency.
Calculate the chi-square statistic for each cell and sum them up.
Speed Never speed Row Total
Male (222-210)²/210 + (78-90)²/90 300
Female (128-140)²/140 + (72-60)²/60 200
Column Total 350 150 500
Summing up the values, we get:
χ2 = [(12²/210) + (12²/90) + (12²/140) + (12²/60)]
Step 4: Determine the critical value and p-value
- The critical value is chosen based on the significance level (1% in this case) and the degrees of freedom.
- The degrees of freedom can be calculated as df = (number of rows - 1) * (number of columns - 1).
In this case, the degrees of freedom are (2-1) * (2-1) = 1 * 1 = 1.
Using a chi-square distribution table or calculator, find the critical value associated with a significance level of 1% and 1 degree of freedom.
The p-value can also be calculated using the chi-square distribution table or calculator. The p-value is the probability of obtaining a test statistic as extreme as the observed chi-square statistic, assuming the null hypothesis is true.
Step 5: Compare the chi-square statistic with the critical value and p-value
- If the chi-square statistic is larger than the critical value or the p-value is smaller than the significance level, we reject the null hypothesis.
- If the chi-square statistic is smaller than the critical value and the p-value is larger than the significance level, we fail to reject the null hypothesis.
Step 6: Interpret the results
- If we reject the null hypothesis, we conclude that there is evidence of a relationship between gender and speeding.
- If we fail to reject the null hypothesis, we conclude that there is not enough evidence to suggest a relationship between gender and speeding.
Hope this explanation helps! Let me know if you have any further questions.