# [[EDUC 7215]] Assignment 8 Jethro Jones also available at [drjethro.com](https://drjethro.com/7215ass8) ## Assignment ### Introduction ### Research Question Is pro-environmental behavior (PEB) a linear function of NEP, PN, and Gender for survey respondents in the Missouri area? #### Type 1 Error Rate 5% or 0.05 to define statistical significance. ### Hypotheses | Intercept | Slope of NEP | Slope of PN | Slope of Gender | | --------------------- | --------------------- | --------------------- | --------------------- | | H₀:  0 = 0 | H₀:  1 = 0 | H₀:  2 = 0 | H₀:  3 = 0 | | H_a:  0 ≠ 0 | H_a:  1 ≠ 0 | H_a:  2 ≠ 0 | H_a:  3 ≠ 0 | ### Methods I clicked Tasks and Utilities, then Tasks, then Linear Models, then Linear Regression. I selected our work set from my work library. Under Roles, I selected PEB for the dependent variable, Gender as the classification variable, and NEP and PN for the continuous variable. Under the Model tab, I selected PN, NEP and Gender and add. In options, I selected Individual plots for both the diagnostic and residual plots. After I clicked Run, I opened it in a new tab and saved the graphs and took screenshots as you'll see below. I used an AI renaming tool to rename the screenshots and graphs appropriately. I then used Tasks and Utilities, then Tasks, then Statistics, then Summary Statistics to generate summary statistics for PEB and PN. Under Option, then Basic Statistics, I check Mean, Standard Deviation, minimum Value, Maximum Value and Median. Under Additional Statistics, I checked 95% Confidence Limits for the mean. Under Plots, I notched the Comaprative box plot. After I clicked Run, I opened it in a new tab and saved the graphs and took screenshots as you'll see below. I used an AI renaming tool to rename the screenshots and graphs appropriately. Gender 0 is male Gender 1 is female Gender 3 is non-binary. We will ignore this as n=6, which is too small for us to pay attention to. ## Summary Statistics The mean PEB scores for males and females is similar (both less than 3), but in all three areas, the female responses are higher than the males, which leads me to believe that gender does have an impact on PEB, NEP, and PN. The confidence intervals show overlap between each score, so gender differences in the MLR model may not be significant. ![[2025-04-02 Statistical Summary Table.png]] ## Box and Whisker Diagrams for these three independent variables by Gender follow ![[2025-04-02 Distribution of PN by gender.png]] ![[2025-04-02 Distribution of NEP by gender.png]] ![[2025-04-02 Distribution of PEB by gender.png]] ## Assumptions of Normality The histogram of the residuals appears to be bell-shaped like we would expect for normally distributed data. The Q-Q plot also shows that residuals follow the solid line, with some minor deviations int he tails. Based on these graphs, I believe the residuals are normally distributed. With a sample size of 379, it is large enough to confirm they are normally distributed. ![[2025-04-02 Distribution of Residuals for PEB.png]] ![[2025-04-02 ResidualFitSpreadPlotPEB.png]] ![[2025-04-02 QQPlotResidualsPEB.png]] ## Assumption of Equal Variance The Studentized residual plot does not show any discernable trands, though there are a few circles that extend beyond +2 and -2 standard deviation lines. However they appear to be evenly distributed across the range of predicted values (on the x axis), so I conclude the variances are equal. ![[2025-04-02 RStudent_vs_Predicted_PEB.png]] ## Assessment of Influential Outliers - Cook's D The Cook's D plot shows a few possible influential outliers, but the range is small. ![[2025-04-02 CooksDPlot.png]] ## Model Fit Statistics The Root MSE=.49209 which is about half a rating. So the model is within one half a rating scale of the true value about 66% of the time. The R-square measure the amount of variability explained by the indenpenedent variables. Since this is an MLR model, we need to use the adjusted R-square to account for two or more independent variables. For this analysis, adjusted R-square = 0.3449 or 34% of the variability caused by NEP, PEB, and PN in gender disparity, which seems quite high. Based on the fit statistics, this is a moderate MLR model with relatively high precision. ![[2025-04-02 Least Squares Model Analysis.png]] ## Parameter hypothesis tests We conducted hypothesis tests for each parameter to determine whether the associated variable has a statistically significant effect on the dependent variable. For the intercept, the null hypothesis stated that the coefficient is equal to zero (<i>β₀ = 0</i>), meaning there is no baseline effect. The p-value for the intercept is 0.0008, which is below the standard significance level (0.05), so we reject the null hypothesis and conclude that the intercept is significant. For NEP and PN, the null hypotheses (<i>β₁ = 0</i> and <i>β₂ = 0</i>, respectively) suggest no effect on the outcome. Both NEP and PN have p-values less than 0.0001, indicating strong evidence to reject the null hypotheses—these variables significantly contribute to the model. In contrast, gender’s coefficients (<i>β₃</i>) have high p-values (0.9505 and 0.7423), so we fail to reject the null hypothesis for gender. This suggests that gender does not have a statistically significant impact on the dependent variable in this model. ## Summary and Conclusion The multiple linear regression model of Pro-Environmental Behavior (PEB) as a linear function of the independent variables NEP, Personal Norms (PN), and Gender (male or female) is statistically significant and demonstrates moderate explanatory power. About 35% of the variability in PEB is explained by the model (adjusted R-square = 34.49%), and the precision is relatively high, with a Root MSE of approximately 0.49—indicating that predicted values are typically within half a rating scale point of the true values. The assumptions of normality and equal variance are satisfied based on residual diagnostics, and there is no indication of multicollinearity among the independent variables. While NEP and PN significantly predict PEB, gender does not have a statistically significant effect in this model. The final model for females and males can be written as: **Female PEB** = 0.901253 + 0.206537·NEP + 0.204698·PN − 0.012843 **Male PEB** = 0.901253 + 0.206537·NEP + 0.204698·PN + 0.067566 Based on average NEP and PN scores by gender, the predicted PEB values are approximately: **Female = 2.84** **Male = 3.00**