Question

In a regression analysis involving 27 observations, the following estimated regression equation was developed: ŷ = 25.2 + 5.5x1 For this estimated regression equation SST = 1,550 and SSE = 520. a. At a = 0.05, test whether x₁ is significant. O F = 49.52; p-value is less than 0.01; x₁ is not significant. F = 46.27; p-value is less than 0.01; x₁ is significant. F = 49.52; critical value is 4.24; x₁ is significant. O F = 51.32; critical value is 4.24; x₁ is significant. Question 21 5 pts b. Suppose that variables x2 and x3 are added to the model and the following regression equation is obtained. ŷ = 16.3 +2.3x₁ + 12.1x2 - 5.8x3 For this estimated regression equation SST = 1,550 and SSE = 100. Use an F test and a 0.05 level of significance to determine whether x2 and x3 contribute significantly to the model. F = 48.3; critical value is 4.28; x2 and x3 contribute significantly to the model. OF = 48.3; p-value is less than 0.01; x2 and x3 contribute significantly to the model. F = 48.3; critical value is 3.42; x2 and x3 don't contribute significantly to the model. O F = 111.17; p-value is less than 0.01; x2 and x3 contribute significantly to the model.

Answer 1

a. To test whether x₁ is significant at a = 0.05, calculate the F-statistic and compare it to the critical value. Reject the null hypothesis if the calculated F-value is greater than the critical value. b. To test whether x2 and x3 contribute significantly to the model at a = 0.05, calculate the F-statistic and compare it to the critical value. Reject the null hypothesis if the calculated F-value is greater than the critical value.

Step-by-step explanation:

a. To test whether x₁ is significant at a = 0.05, we can calculate the F-statistic using the formula F = (SSR / k) / (SSE / (n - k - 1)).
Here, SSR = SST - SSE = 1550 - 520 = 1030, k = 1 (number of regressors), SSE = 520, and n = 27 (number of observations).
Plugging in these values, we get F = (1030 / 1) / (520 / (27 - 1 - 1)) = 49.52.
Next, we can compare this value to the critical value from the F-distribution table with (k, n - k - 1) degrees of freedom. At a = 0.05, the critical value is 4.24 for (1, 25) degrees of freedom.
Since the calculated F-value of 49.52 is greater than the critical value of 4.24, we reject the null hypothesis. Therefore, x₁ is significant.

b. To test whether x2 and x3 contribute significantly to the model at a = 0.05, we can calculate the F-statistic using the formula F = ((SSE₁ - SSE₂) / (k₂ - k₁)) / (SSE₂ / (n - k₂ - 1)).
Here, SSE₁ = 520, SSE₂ = 100, k₁ = 1 (number of regressors in the initial model), k₂ = 3 (number of regressors in the new model), and n = 27 (number of observations).
Plugging in these values, we get F = ((520 - 100) / (3 - 1)) / (100 / (27 - 3 - 1)) = 48.3.
Next, we can compare this value to the critical value from the F-distribution table with ((k₂ - k₁), (n - k₂ - 1)) degrees of freedom. At a = 0.05, the critical value is 4.28 for (2, 23) degrees of freedom.
Since the calculated F-value of 48.3 is greater than the critical value of 4.28, we reject the null hypothesis. Therefore, x2 and x3 contribute significantly to the model.

Answer 2

In statistics, the significance of variables in a regression model is determined using F tests, by comparing the calculated F statistic to critical values or examining the p-value in relation to the significance level.

Step-by-step explanation:

The subject question involves regression analysis and hypothesis testing in statistics, specifically the use of F tests to determine the significance of variables in a regression model. Regarding part a, with SST (Total Sum of Squares) = 1550 and SSE (Error Sum of Squares) = 520 for 27 observations, we calculate SSR (Regression Sum of Squares) as SST - SSE, and then compute the F statistic for the regression as (SSR/1) / (SSE/(n-2)), where n is the number of observations. To test the significance of x1, we look at the p-value and compare it with the significance level α = 0.05. If the p-value is less than α, we reject the null hypothesis, concluding that x1 is a significant predictor. For part b, the addition of x2 and x3 has reduced SSE to 100 while SST remains 1550. An F test determines if these new variables contribute significantly to the model. Again, we calculate the F statistic and compare it to a critical value for the appropriate degrees of freedom at the α significance level or look at the p-value.