Question

The maintenance manager at J.B. Inc. believes that she can use regression analysis to predict the weekly maintenance cost based on the number of hours a machine operates per week. Data from the last nine weeks are presented below.

Σ X = 150

Σ XY = 4649

Σ X2 = 2595

Σ Y2 = 8730

Cost (Y) $000’s: 44, 22, 41, 15, 14, 35, 30, 39, 21

At the 0.01 level of significance, can we conclude that a significant positive relationship exists between maintenance cost and the number of hours a machine operates per week? Comment on the necessity of the independent variable in this regression model. Show your calculations and use the p-value approach.

Answer 1

Solution :

The null and alternative hypotheses would be as follows :

We shall use t-test for significance of correlation coefficient. The test statistic is given as follows :

Where, r is sample correlation coefficient and n is sample size.

We have, n = 9,

The value of the test statistic is 5.4728.

Degrees of freedom = (n - 2) = (9 - 2) = 7

Our test is right-tailed test. The right-tailed p-value is given as follows :

P-value = P(T > t)

P-value = P(T > 5.4728)

P-value = 0.0005

The p-value is 0.0005.

Significance level = 0.01

(0.0005 < 0.01)

Since, p-value is less than the significance level of 0.01, there we shall reject the null hypothesis (H_{​​​​​​0}) at 0.01 significance level.

Conclusion : At 0.01 significance level, there is sufficient evidence to conclude that a significant positive relationship exists between maintenance cost and the number of hours a machine operates per week.

One of the assumption of the regression model is that observations must be independent.