Please clarify the answer mentioned below the following question. I know the formula used for the t-test (t = (^b1 - b1) / S^b1) but I’m confused about how they came up with b1 numbers (6 for Hypothesis 1, and -1 for Hypothesis 2)?!
Question ID: 1074723
Theresa Miller is attempting to forecast sales for Alton Industries based on a multiple regression model. The model Miller estimates is:
sales = b0 + (b1 × DOL) + (b2 × IP) + (b3 × GDP) +εt where: sales = change in sales adjusted for inflation DOL = change in the real value of the (rates measured in €/) IP = change in industrial production adjusted for inflation (millions of ) GDP = change in inflation-adjusted GDP (millions of )
All changes in variables are in percentage terms.
Miller runs the regression using monthly data for the past 180 months. The model estimates (with coefficient standard errors in parentheses) are:
sales = 10.2 + (5.6 × DOL) + (6.3 × IP) + (9.2 × GDP) (5.4) (3.5) (4.2) (5.3)
Miller tests and fails to reject each of the following two null hypotheses at the 99% confidence interval:
Hypothesis 1: A 2% increase in DOL will result in an increase in sales of more than 12%. Hypothesis 2: A 1% increase in industrial production will result in a 1% decrease in sales.
LOOK UP: >> Partial Table of Student’s t-Distribution (One-Tailed Probabilities) <<
Did Miller correctly interpret the results of the tests in making her reject or fail-to-reject decisions for Hypothesis 1 and Hypothesis 2?
Hypothesis 1 Hypothesis 2
A)
Yes Yes
The critical values at the 1% level of significance (99% confidence) are 2.348 for a one-tailed test and 2.604 for a two-tailed test (df = 176).
Hypothesis 1: This hypothesis is asking whether a 2% increase in DOL will increase sales by more than 12%. This will only happen if the value of the coefficient is greater than 6, since 2 × 6 = 12. Since the regression estimate for this coefficient is 5.6, the t-statistic for this test is (5.6 – 6) / 3.5 = –0.114. This is a one-tailed test, so the critical value is 2.348. Miller is correct in failing to reject the null.
Hypothesis 2: This hypothesis is asking whether the value of the coefficient is equal to –1.0, since that is the value that would correspond with a 1% increase in industrial production, resulting in a 1% decrease in sales. Since the regression estimate for this coefficient is 6.3, the t-statistic for this test is [6.3 – (–1)] / 4.2 = 1.74. This is a two-tailed test, so the critical value is 2.604. Miller is correct in failing to reject the null. ((Study Session 3, Module 8.2, LOS 8.d))