statistic question

Question 63

· Conditions Check:

· Income scatterplot: This scatterplot has a positive direction with a weak linear pattern.GPA scatterplot: This scatterplot has a positive direction with a strong linear pattern.

· A:

· Question tells us to estimate:

· Model 1: = 𝞫₀ + 𝜷₁ Income + ε

· Model 2: = 𝞫₀ + 𝜷₁ GPA + ε

· Model 3: = 𝞫₀ + 𝜷₁ Income + 𝜷₂ GPA + ε

· The Multiple Linear Regression Model has:

· Explanatory Variables: Income, GPA

· Response Variable: SAT

· Scatterplots:

Source: sampling data from Jaggie chapter 14, question 63, downloaded March 26, 2019.

Source: sampling data from Jaggie chapter 14, question 63, downloaded March 26, 2019.

· Equations:

· Model 1: = 𝞫₀ + 𝜷₁ Income + ε

1. = 1616.4 + 0.001470 Income

1. Model 2: = 𝞫₀ + 𝜷₁ GPA + ε

2. = 1259.6 + 141.5 GPA

1. Model 3: T = 𝞫₀ + 𝜷₁ Income + 𝜷₂ GPA + ε

3. = 1104.3 + 151.0 GPA + 0.001705 Income

· B: Goodness of fit measures, sample size 24

The best-fitting model is Model 3 (SAT = 𝞫₀ + 𝜷₁ Income + 𝜷₂ GPA + ε) because it has the highest adjusted R² of 85.20%.

C: = 1104.3 + (0.001705 * 72833.3) + (151.0 * 3.278333) = 1723.509

D: Interpretation: Holding constant income and GPA, the predicted number of SAT increases by 151.0 and 0.001705, respectively. The estimated sample regression equation shown in model 3 explains approximately 86.49% of the sample variability in the SAT.

Appendix:

Table of sample statistics:

Summary Output:

· SAT vs Income

· SAT vs GPA

· SAT vs GPA, Income

· Histograms

· Scatterplots