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 = 1,500 and SSE = 590.(a)At α = 0.05, test whether x₁ is significant. State the null and alternative hypotheses.

H₀: β1 ≠ 0
Ha: β₁ = 0 H₀: β₁ = 0
Ha: β₁ ≠ 0 H₀: β₀ ≠ 0
Ha: β₀ = 0H₀: β₀ = 0
Ha: β₀ ≠ 0
Find the value of the test statistic. (Round your answer to two decimal places.)F =_____ Find the p-value. (Round your answer to three decimal places.)
p-value =______ Is x1 significant? Do not reject H₀. We conclude that x₁ is not significant.Reject H₀. We conclude that x₁ is not significant. Reject H₀. We conclude that x₁ is significant.Do not reject H₀. We conclude that x1 is significant. Suppose that variables x₂ and x₃ are added to the model and the following regression equation is obtained. ŷ = 16.3 + 2.3x₁ + 12.1₂ − 5.8x₃ For this estimated regression equation SST = 1,500 and SSE = 100. (b) Use an F test and a 0.05 level of significance to determine whether x₂ and x₃ contribute significantly to the model. State the null and alternative hypotheses. H₀: One or more of the parameters is not equal to zero. Ha: β₂ = β₃ = 0 H₀: β₂ = β₃ = 0 Ha: One or more of the parameters is not equal to zero. H₀: β₁ ≠ 0 Ha: β₁ = 0 H0: β₁ = 0 Ha: β₁ ≠ 0 Find the value of the test statistic. Find the p-value. (Round your answer to three decimal places.) p-value = Is the addition of x₂ and x₃ significant?
Do not reject H0. We conclude that the addition of variables x₂ and x₃ is not significant.Reject H₀. We conclude that the addition of variables x₂ and x₃ is not significant. Do not reject H₀. We conclude that the addition of variables x₂ and x₃ is significant.Reject H₀. We conclude that the addition of variables x₂ and x₃ is significant.

Answer 1

Using the F test and a significance level of 0.05, we determine whether x1 and x2 contribute significantly to the regression model.

Step-by-step explanation:

To test whether x1 is significant, we need to use the F test and a significance level of 0.05. The null hypothesis (H0) states that β1 = 0, and the alternative hypothesis (Ha) states that β1 ≠ 0. The F statistic is calculated by dividing the mean squared regression (MSR) by the mean squared error (MSE). In this case, the F statistic is 1500/590 = 2.54. The p-value is then determined from the F distribution with the appropriate degrees of freedom (1, 25) and is found to be 0.126. Since the p-value > 0.05, we fail to reject the null hypothesis. Therefore, we conclude that x1 is not significant in the regression model.

To test whether x2 and x3 contribute significantly to the model, we use the same F test and significance level of 0.05. The null hypothesis (H0) states that β2 = β3 = 0, and the alternative hypothesis (Ha) states that one or more of the parameters is not equal to zero. The F statistic is calculated by dividing the mean squared regression (MSR) by the mean squared error (MSE).

In this case, the F statistic is 1500/100 = 15. The p-value is then determined from the F distribution with the appropriate degrees of freedom (2, 21) and is found to be less than 0.001. Since the p-value < 0.05, we reject the null hypothesis. Therefore, we conclude that the addition of x2 and x3 is significant in the regression model.