To test for the significance of the regression, we typically perform a regression analysis and then conduct an F-test to determine if the regression model is a good fit for the data. We will compare the obtained F-statistic to the critical F-value from the F-distribution table based on the degrees of freedom.
Step 1: Calculate the necessary components for regression analysis
- Calculate means of X and Y.
- Calculate the regression coefficients (slope \(b\) and intercept \(a\)) using least squares method.
- Calculate the Total Sum of Squares (SST), Regression Sum of Squares (SSR), and Residual Sum of Squares (SSE).
- Calculate the F-statistic.
Step 2: Determine the degrees of freedom
- \( k = 1 \): Number of predictors (in this case, only Y)
- \( n = 10 \): Total number of observations (10 students)
- So, the degrees of freedom for the regression is \( df_{regression} = k = 1 \) and for the residual is \( df_{residual} = n - k - 1 = 10 - 1 - 1 = 8 \).
Step 3: Look up F-table value
For a significance level of \( \alpha = 0.01 \) with \( df_{regression} = 1 \) and \( df_{residual} = 8 \), you can consult an F-distribution table:
- The critical value \( F(1, 8) \) at \( \alpha = 0.01 \) is approximately 8.612.
Conclusion
If the F-statistic computed from your regression analysis is greater than 8.612, you would reject the null hypothesis and conclude that there is a statistically significant relationship between the test scores (X) and the final examination scores (Y) at the \( \alpha = 0.01 \) significance level. If it is less, you would fail to reject the null hypothesis.
If you can perform the regression analysis calculations, you will find the F-statistic that you can compare with this critical value.
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