A forecaster used the regression equation $Q _ { t } = a + b t + c _ { 1 } D _ { 1 } + c _ { 2 } D _ { 2 } + c _ { 3 } D _ { 3 }$ and quarterly sales data for 2004I-2021IV (t = 1,...,64)for an appliance manufacturer to obtain the results shown below.Q is quarterly sales,and $D _ { 1 } , D _ { 2 }$ and $D _ { 3 }$ are dummy variables for quarters I,II,and III. At the 5 percent level of significance,is there a statistically significant trend in sales? A)No,because 1.5

Question

A forecaster used the regression equation $Q _ { t } = a + b t + c _ { 1 } D _ { 1 } + c _ { 2 } D _ { 2 } + c _ { 3 } D _ { 3 }$ and quarterly sales data for 2004I-2021IV (t = 1,...,64)for an appliance manufacturer to obtain the results shown below.Q is quarterly sales,and $D _ { 1 } , D _ { 2 }$ and $D _ { 3 }$ are dummy variables for quarters I,II,and III.  At the 5 percent level of significance,is there a statistically significant trend in sales?
A)No,because 1.5 < 2.66. 
B)No,because 1.5 < 2.00. 
C)No,because 2.14 < 2.66. 
D)Yes,because 2.14 > 2.00. 
E)none of the above

Quizplus · Accepted Answer

To test whether there is a statistically significant trend in sales, we need to test the null hypothesis that the slope coefficient (b) is equal to zero (i.e., there is no trend) versus the alternative hypothesis that the slope coefficient is not equal to zero (i.e., there is a trend). The t-statistic for testing this hypothesis is given by t = b/se(b), where se(b) is the standard error of the slope coefficient. From the regression output, we have b = 1.5 and se(b) = 0.704. The t-statistic is therefore t = 1.5/0.704 = 2.14. The critical t-value for a two-tailed test at the 5% level of significance with 61 degrees of freedom (64 - 3 dummy variables - 1 intercept) is approximately 2.00. Since the computed t-statistic (2.14) is greater than the critical t-value (2.00), we reject the null hypothesis and conclude that there is a statistically significant trend in sales. Therefore, the correct choice is D, Yes, because 2.14 > 2.00.

A Forecaster Used the Regression Equation Qt=a+bt+c1D1+c2D2+c3D3Q _ { t } = a + b t + c _ { 1 } D _ { 1 } + c _ { 2 } D _ { 2 } + c _ { 3 } D _ { 3 }Qt​=a+bt+c1​D1​+c2​D2​+c3​D3​

A Forecaster Used the Regression Equation $Q _ { t } = a + b t + c _ { 1 } D _ { 1 } + c _ { 2 } D _ { 2 } + c _ { 3 } D _ { 3 }$