# Linear Regression Solution Example

Linear Regression Analysis

The independent variable selected for this analysis is AGE, and the independent variable selected for this analysis is HRS1 (hours worked last week). Both of these variables are measured at the ratio level. The main purpose of the analysis is to assess whether or not these two variables have a significant degree of linear association. Chart 1 below exhibits a scatterplot for these two variables.

Chart 1: Scatterplot of HRS1 versus AGE

The scatterplot shown in Chart 1 indicates that a very diffuse association is exhibited, and it possibly indicates a very weak negative association.

## The Null and Alternative Hypothesis

The following needs to be tested.

Ho: $$\rho = 0$$

Ha: $$\rho \ne 0$$

The assumption for testing the above hypotheses is that both AGE and HRS1 are normally distributed. Table 1 below shows the correlation between AGE and HRS1.

 Correlations AGE OF RESPONDENT NUMBER OF HOURS WORKED LAST WEEK AGE OF RESPONDENT Pearson Correlation 1 -.325** Sig. (2-tailed) .000 N 1495 1485 NUMBER OF HOURS WORKED LAST WEEK Pearson Correlation -.325** 1 Sig. (2-tailed) .000 N 1485 1490 **. Correlation is significant at the 0.01 level (2-tailed).

The correlation for AGE and HRS1 is r = -.325, p < .001, which is significantly different from zero. Therefore, the null hypothesis Ho is rejected. Furthermore, there is enough evidence to claim that there is a significant, negative association between the number of hours worked last week and age. However, the strength of this negative association is rather weak..

Tables 2, 3 and 4 below show the regression results.

 Table 2: Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate 1 .325a .105 .105 22.302 a. Predictors: (Constant), AGE OF RESPONDENT
 Table 3: ANOVA Model Sum of Squares df Mean Square F Sig. 1 Regression 86941.814 1 86941.814 174.798 .000b Residual 737619.214 1483 497.383 Total 824561.028 1484 a. Dependent Variable: NUMBER OF HOURS WORKED LAST WEEK b. Predictors: (Constant), AGE OF RESPONDENT
 Table 4: Regression Coefficients Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) 48.162 1.704 28.267 .000 AGE OF RESPONDENT -.458 .035 -.325 -13.221 .000 a. Dependent Variable: NUMBER OF HOURS WORKED LAST WEEK

The regression model is significant overall, F(1, 1483) = 174.798, p < .001. The model is

HRS1 = 48.162 – 0.458*AGE

This model explains only 10.5% of the variation in HRS1. The model indicates that for extra year of age, the number of hours worked decreases by 0.458 hours, on average.

# SPSS Syntax

GRAPH

/SCATTERPLOT(BIVAR)=AGE WITH HRS1

/MISSING=LISTWISE.

CORRELATIONS

/VARIABLES=AGE EDUC HRS1

/PRINT=TWOTAIL NOSIG

/MISSING=PAIRWISE.

# References

Gravetter, F. & Wallnau, L. (2013). Essentials of Statistics for the Behavioral Sciences. Wadsworth Publishing; 8th edition.