Question

Please assist with the following question:

5.

A simple linear regression equation relates V and S as follows:

V = a + b*S   

The linear regression in problem 1 is estimated using 62 observations on V and S. The least-squares estimate of b is –21.749, and the standard error of the estimate is 9.10. Perform a t-test for statistical significance at the 5 percent level of significance

a. There are ______degrees of freedom for the t-test.

b. The value of the t-statistic is ______. The critical t-value for the test is ______.

c. Is statistically significant? Explain.

d. The p-value for the t-statistic is ______. (Hint: In this problem, the t-table provides the answer.) The p-value gives the probability of rejecting the hypothesis that ______(b = 0, b > 0, b < 0) when b is truly equal to ______. The confidence level for the test is ______ percent.

e. What does it mean to say is statistically significant at the 5 percent significance level?

f. What does it mean to say is statistically significant at the 95 percent confidence level?

g. Explain the difference between the level of confidence differ from the level of significance?

Answer 1

Solution:-

5)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

H₀: The slope of the regression line is equal to zero.
H_a: The slope of the regression line is not equal to zero.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a linear regression t-test to determine whether the slope of the regression line differs significantly from zero.

Analyze sample data. To apply the linear regression t-test to sample data, we require thestandard error of the slope, the slope of the regression line, the degrees of freedom, the t statistic test statistic, and the P-value of the test statistic.

We get the slope (b₁) and the standard error (SE) from the regression output.

b₁ = - 21.749 SE = 9.10

We compute the degrees of freedom and the t statistic test statistic, using the following equations.

a)

DF = n - 2

D.F = 60

b)

t = b₁/SE

t = - 2.39

t_critical = + 2.0

Rejection region is - 2.0 > t > + 2.0

where DF is the degrees of freedom, n is the number of observations in the sample, b₁ is the slope of the regression line, and SE is the standard error of the slope.

c) Yes, the results are statistically significant, because t-values lies in rejection region.

d)

Therefore, the P-value is 0.02.

Interpret results. Since the P-value (0.02) is less than the significance level (0.05), we cannot accept the null hypothesis.

The p-value gives the probability of rejecting the hypothesis that b = 0 when b is truly equal to 0. The confidence level for the test is 95 percent.