Step 2 MSR The F test statistic is calculated as F =. where MSR is the mean square due to regression and MSE is the MSE SSR mean square error. The MSR is calculated as MSR = where SSR is the sum of squares due to regression p and p is the number of independent variables in the estimated regression equation. The MSE is calculated as SSE MSE = where SSE is the sum of square error, n is the total number of observations used to create n- p – 1 the estimated regression equation, and p is again the number of independent variables. The values for SST and SSE are given, so SSR must be found. Recall the relationship between the sum of squares. SST = SSR + SSE We are given SST = 1,525 and SSE = 580. Therefore, we can find SSR as follows. SSR = SST – SSE = 1,525 – 580 The estimated regression equation given below was based on 27 observations. ý = 25.2 + 5.5x The estimated regression equation has Since there were 27 observations, we have n = independent variable(s), so p =
The Correct Answer and Explanation is:
To solve this problem step by step and calculate the F-test statistic for the regression, we need to follow these procedures:
Step 1: Find SSR (Sum of Squares due to Regression)
We are given:
- Total Sum of Squares (SST) = 1525
- Sum of Squares Error (SSE) = 580
Using the identity: SSR=SST−SSE=1525−580=945SSR = SST – SSE = 1525 – 580 = 945
Step 2: Determine p and n
- The estimated regression equation is:
y^=25.2+5.5x\hat{y} = 25.2 + 5.5x
This has one independent variable, so p=1p = 1.
- The number of observations is given as 27, so n=27n = 27
Step 3: Calculate MSR and MSE
MSR (Mean Square Regression):
MSR=SSRp=9451=945MSR = \frac{SSR}{p} = \frac{945}{1} = 945
MSE (Mean Square Error):
MSE=SSEn−p−1=58027−1−1=58025=23.2MSE = \frac{SSE}{n – p – 1} = \frac{580}{27 – 1 – 1} = \frac{580}{25} = 23.2
Step 4: Calculate the F-Statistic
F=MSRMSE=94523.2≈40.73F = \frac{MSR}{MSE} = \frac{945}{23.2} \approx 40.73
✅ Final Answer:
- F = 40.73
📘 Explanation
In linear regression, the F-test statistic is used to determine whether the regression model provides a better fit to the data than a model with no predictors. It compares the model’s explained variance to the unexplained variance. This comparison is made through the ratio of two quantities: the Mean Square due to Regression (MSR) and the Mean Square Error (MSE).
MSR measures the average variation explained by the model per independent variable, and MSE measures the average variation in the residuals (unexplained part) per degree of freedom left after fitting the model.
To compute the F-statistic:
- We first find the SSR by subtracting the SSE (unexplained variation) from SST (total variation). This gives us the variation explained by the regression model.
- We then divide SSR by the number of predictors (p) to get MSR.
- Similarly, we divide SSE by the degrees of freedom left after fitting the model (n − p − 1) to get MSE.
- Finally, dividing MSR by MSE gives the F-statistic, which tells us whether the explained variance is significantly greater than the unexplained variance.
In our case, a large F-value of 40.73 suggests that the regression model explains a significant portion of the variance in the response variable, indicating that the independent variable xx contributes meaningfully to the model. This is a strong indication that the predictor has a statistically significant relationship with the response variable.
