Sixty-four students in an introductory college economics class were askd how many credits they had earned in college, and how certain they were about their choice of major. At a = .01, is the degree of certainty independent of credits earned?
Credits Very Somewhat Very Row
Earned uncertain certain certain total
0-9 12 8 3 23
10-59 8 4 10 22
60 ^ 1 7 11 19
Col total 21 19 24 64
Sixty-four students in an introductory college economics class were askd how many credits they had earned in college, and how certain they were about their choice of major. At a = .01, is the degree of certainty independent of credits earned?
Credits Very Somewhat Very Row
Earned uncertain certain certain total
0-9 12 8 3 23
10-59 8 4 10 22
60 ^ 1 7 11 19
Col total 21 19 24 64
To determine if the degree of certainty is independent of credits earned at a significance level of α = 0.01, we need to perform a chi-square test of independence.
Step 1: Set up hypotheses
- Null Hypothesis (H0): The degree of certainty is independent of credits earned.
- Alternative Hypothesis (Ha): The degree of certainty is dependent on credits earned.
Step 2: Define the significance level
Given that a = 0.01, we have α = 0.01.
Step 3: Calculate the expected frequencies
To perform the chi-square test of independence, we need to calculate the expected frequencies for each cell. The expected frequency for each cell can be calculated using the formula:
Expected Frequency = (Row Total * Column Total) / Grand Total
Using the given data, we can calculate the expected frequencies for each cell:
Expected Frequency for "Very uncertain" in the "0-9 credits earned" category: (23 * 21) / 64 = 7.234375
Expected Frequency for "Somewhat certain" in the "0-9 credits earned" category: (23 * 19) / 64 = 6.796875
Expected Frequency for "Very certain" in the "0-9 credits earned" category: (23 * 24) / 64 = 8.53125
Similarly, calculate the expected frequencies for the remaining cells.
Step 4: Compute the chi-square test statistic
The chi-square test statistic can be calculated using the formula:
χ^2 = Σ ( (Observed Frequency - Expected Frequency)^2 / Expected Frequency )
Calculate the contribution to the chi-square test statistic for each cell and sum them to get the final test statistic.
Step 5: Determine the critical value
Since the degrees of freedom (df) for a chi-square test of independence are calculated as (number of rows - 1) * (number of columns - 1), we have df = (3-1) * (3-1) = 4.
Using a chi-square table or a calculator, find the critical value for α = 0.01 and df = 4. Let's assume the critical value is C.
Step 6: Make a decision
If the calculated test statistic (χ^2) is greater than the critical value (C), then we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 7: Conclusion
Based on the decision made in Step 6, we conclude whether the degree of certainty is independent of credits earned at a significance level of α = 0.01.
Perform these calculations to get the final result.