Question

Managers are an important part of any organization’s resource base. Accordingly, the organization should be just as concerned about forecasting its future managerial needs as it is with forecasting its needs for, say, the natural resources used in its production process. A common forecasting procedure is to model the relationship between sales and the number of managers needed, since the demand for managers is the result of the increases and decreases in the demand for products and services that a firm offers its customers. To develop, this relationship data were collected from a firm’s records. An Excel printout of the simple linear regression is provided below.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.963863
R Square	0.929031
Adjusted R Square	0.925088
Standard Error	2.566419
Observations	20
ANOVA
	df	SS	MS	F	Significance F
Regression	1	1551.993	1551.993	235.6323	8.75E-12
Residual	18	118.5571	6.586505
Total	19	1670.55
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	5.325299	1.179868	4.513471	0.000269	2.846489	7.804109	2.846489	7.804109
Sold	0.5861	0.038182	15.35032	8.75E-12	0.505883	0.666317	0.505883	0.666317

1. What is the equation of the sample regression line?
2. Test the intercept and slope estimates for statistical significance at the 1 percent significance level. Explain how you performed this test, and present your results.
3. Interpret the p-values for the parameter estimates.
4. Test the overall equation for statistical significance at the 1 percent significance level. Explain how you performed this test, and present your results. Interpret the p-value for the F-statistic.
5. If the value of sold is 24, what is the fitted (or predicted) value of managers?

Answer 1

The equation of the sample regression line is:

Number of managers= 5.325299+(0.5861*Sold)

Here number of managers is the dependent (y) variable, 5.325299 is the intercept term and Sold is the independent (x) variable. 0.5861 is the coefficient of the independent variable under question.

We look at the p-values to judge whether the intercept and independent variable is significant at the 1% level of significance or not.

The null hypothesis for testing for significance of individual parameters (intercept and sold in this case) is

H0: the parameter = 0 implying that the parameter is insignificant

Against

H1: the parameter is not equal to 0, implying that it is significant

For the intercept we find that the p value= 0.000269, which is < than 0.01

As per convention, if the p-value is greater than the level of significance (alpha), we accept the null hypothesis

Otherwise we reject it.

So, the intercept's p-value is less than 0.01 (alpha)

We reject the null hypothesis at 1% level of significance.

Therefore, we can say that the intercept is a significant parameter and has a significant impact on the dependent variable (the number of managers needed)

For the independent variable (sold) also we come to the same conclusion by looking at the corresponding p-value (8.75E-12) which is far less than the level of significance of 0.01.

So, sold also significantly affects number of managers.

Note: We can also look at the t-values and compare the calculated t-values with the critical t-values present in the t-table. However, it's always preferable that we look at the p-values instead and derive the solutions. Hence that's what I've done.

Just to give a brief as to how t is calculated for the intercept and sold:

t= coefficients/ standard error of the coefficient

Coming to the second part of the question, this question is about the overall significance of the model. To check for the significance level we must consider the ANOVA table's results.

The null hypothesis for overall significance of the model is

H0: All the parameters= 0

Against the alternative

H1: At least one parameter is not 0

Acceptance of the null hypothesis implies that the model as a whole is insignificant and no variables significantly affect the dependent variable

Acceptance of the alternative hypothesis on the other hand implies that the model is significant and at least one of the parameter has a significant impact on number of managers (dependent variable)

We notice that significance F has a value of 8.75E-12, which is less than 0.01.

Hence we reject the null hypothesis and conclude that the model is significant at 1% level of significance.

Note: We can also use the F value to decide about the significance of the model.

We need to compare calculated F with the critical value of F present in the F table but, a p-value is always preferred over this technique and hence that's what I've done

F= Mean square between / Mean square within

Or, F = (Sum of Squares Regression (SSR)/ degrees of freedom regression (df regression))/ (Sum of squares residual/ degrees of freedom residual)

The third part of the question demands finding the fitted value of number of managers when the value of sold is 24.

Substituting these values in the first equation, we can derive the value.

Number of managers (fitted) = 5.325299+(0.5861*24)

= 5.325299+14.006400

= 19. 331699

= 19 (approximately).