Software Lab 6.2 Solutions

Iain Pardoe

Lesson 6.2: Simple Linear Regression

Software Lab 6.2 Solutions

Personal freedom scores range from 2.17 to 9.40 with a mean of 6.98 and a standard deviation of 1.49. Media control ratings range from 0.75 to 9.75 with a mean of 5.02 and a standard deviation of 2.32. Security and safety ratings range from 3.96 to 9.96 with a mean of 8.07 and a standard deviation of 1.42.

Figure 1: Descriptives table in jamovi for personal freedom data

Personal freedom scores (Fig. 2) are slightly right-skewed, with the majority of scores between about 5 and 9.5.

Figure 2: Personal freedom scores (pf_score)

Media control ratings (Fig. 3) are reasonably symmetric, with the majority of ratings between about 2.5 and 7.5.

Figure 3: Media control ratings

Security and safety ratings (Fig. 4) are somewhat right-skewed, with the majority of ratings between about 7 and 10.

Figure 4: Security and safety ratings
There is a strong negative linear trend (Fig. 5) as the average value of pf_score tends to decrease as pf_media_control increases. There are three points with very low values of pf_score and high values of pf_media_control that stick out slightly from the overall point cloud.

Figure 5: Scatterplot of pf_media_control vs pf_score data
The linear regression line (Fig. 6) has a negative slope and is quite steep. There is a relatively large decrease in average pf_score for each unit increase in pf_media_control.

Figure 6: Scatterplot with negative-slope regression line for pf_score vs pf_media_control data
There is no apparent pattern in the residuals plot (Fig. 7), which indicates it is reasonable to assume a linear relationship between the two variables.

Figure 7: Scatterplot of residuals vs. pf_media_control data
The majority of the points (Fig. 8) lie close to the diagonal line in the normal probability plot with no extreme outliers, which indicates that the nearly normal residuals condition is not violated.

Figure 8: Normal probability plot for pf_score vs. pf_media_control data
The variability of the residuals in the “Residuals vs pf_media_control” plot in question 4 (see Fig. 7) appear reasonably constant across the plot, which indicates that the constant variability condition is not violated. The slight decreases in residual variability for low values of pf_media_control on the left of the plot is not extreme enough to be concerned about.
There is a fairly strong positive linear trend as the average value of pf_score tends to increase as pf_security_safety increases (Fig. 9). The linear regression line has a positive slope and is fairly steep (average pf_score increases a fair amount for each unit increase in pf_security_safety).

Figure 9: Scatterplot with positive-slope regression line for pf_score vs. pf_security_safety data
$\widehat{\text{pf\_score}}=0.627+0.788\,\text{pf\_security\_safety}$ .

Figure 10: Model coefficients table for pf_score data
56.7% of the variability in pf_score is explained by pf_media_control.

Figure 11: Model fit measures table for pf_score vs. pf_security_safety data
The predictor pf_media_control should produce more accurate predictions of pf_score on average because there is a stronger linear association between pf_score and pf_media_control (see question 3) than between pf_score and pf_security_safety (see question 7) and because the value of R² is higher for the linear regression model with pf_media_control (71.4%) than the linear regression model with pf_security_safety (56.7%).