'data.frame': 29 obs. of 5 variables:
$ YEAR : int 1 2 3 4 5 6 7 8 9 10 ...
$ ROLL : int 5501 5945 6629 7556 8716 9369 9920 10167 11084 12504 ...
$ UNEM : num 8.1 7 7.3 7.5 7 6.4 6.5 6.4 6.3 7.7 ...
$ HGRAD: int 9552 9680 9731 11666 14675 15265 15484 15723 16501 16890 ...
$ INC : int 1923 1961 1979 2030 2112 2192 2235 2351 2411 2475 ...
summary(UNM)
YEAR ROLL UNEM HGRAD INC
Min. : 1 Min. : 5501 Min. : 5.700 Min. : 9552 Min. :1923
1st Qu.: 8 1st Qu.:10167 1st Qu.: 7.000 1st Qu.:15723 1st Qu.:2351
Median :15 Median :14395 Median : 7.500 Median :17203 Median :2863
Mean :15 Mean :12707 Mean : 7.717 Mean :16528 Mean :2729
3rd Qu.:22 3rd Qu.:14969 3rd Qu.: 8.200 3rd Qu.:18266 3rd Qu.:3127
Max. :29 Max. :16081 Max. :10.100 Max. :19800 Max. :3345
Simple Plots
Roll vs. Year
ggplot(UNM, aes(x = ROLL, y = YEAR)) +geom_point(color ="#ba0c2f")
Roll vs. January Unemployment
ggplot(UNM, aes(x = ROLL, y = UNEM)) +geom_point(color ="#ba0c2f")
Roll vs. Spring High School Graduates
ggplot(UNM, aes(x = ROLL, y = HGRAD)) +geom_point(color ="#ba0c2f")
Roll vs. Per Captia Income in Albuquerque
ggplot(UNM, aes(x = ROLL, y = INC)) +geom_point(color ="#ba0c2f")
Model
Build the Model
fall =lm(ROLL ~ UNEM + HGRAD, UNM)
Analyze the Model
summary(fall)
Call:
lm(formula = ROLL ~ UNEM + HGRAD, data = UNM)
Residuals:
Min 1Q Median 3Q Max
-2102.2 -861.6 -349.4 374.5 3603.5
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -8.256e+03 2.052e+03 -4.023 0.00044 ***
UNEM 6.983e+02 2.244e+02 3.111 0.00449 **
HGRAD 9.423e-01 8.613e-02 10.941 3.16e-11 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1313 on 26 degrees of freedom
Multiple R-squared: 0.8489, Adjusted R-squared: 0.8373
F-statistic: 73.03 on 2 and 26 DF, p-value: 2.144e-11
anova(fall)
Analysis of Variance Table
Response: ROLL
Df Sum Sq Mean Sq F value Pr(>F)
UNEM 1 45407767 45407767 26.349 2.366e-05 ***
HGRAD 1 206279143 206279143 119.701 3.157e-11 ***
Residuals 26 44805568 1723291
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
High school graduation is more closely related to undergradute enrollment than unenployment rate. This is because the F statisitc of ~ 120. While both varribles were significant, HGRAD has a lower value of 3.157e-11.
Expected enrollment is around 21,586 students. ## New Model ### Build the Model
fall2 =lm(ROLL ~ UNEM + HGRAD + INC, UNM)
Analyze the Model
summary(fall2)
Call:
lm(formula = ROLL ~ UNEM + HGRAD + INC, data = UNM)
Residuals:
Min 1Q Median 3Q Max
-1148.84 -489.71 -1.88 387.40 1425.75
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -9.153e+03 1.053e+03 -8.691 5.02e-09 ***
UNEM 4.501e+02 1.182e+02 3.809 0.000807 ***
HGRAD 4.065e-01 7.602e-02 5.347 1.52e-05 ***
INC 4.275e+00 4.947e-01 8.642 5.59e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 670.4 on 25 degrees of freedom
Multiple R-squared: 0.9621, Adjusted R-squared: 0.9576
F-statistic: 211.5 on 3 and 25 DF, p-value: < 2.2e-16
anova(fall2)
Analysis of Variance Table
Response: ROLL
Df Sum Sq Mean Sq F value Pr(>F)
UNEM 1 45407767 45407767 101.02 2.894e-10 ***
HGRAD 1 206279143 206279143 458.92 < 2.2e-16 ***
INC 1 33568255 33568255 74.68 5.594e-09 ***
Residuals 25 11237313 449493
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Comparing Models
anova(fall, fall2)
Analysis of Variance Table
Model 1: ROLL ~ UNEM + HGRAD
Model 2: ROLL ~ UNEM + HGRAD + INC
Res.Df RSS Df Sum of Sq F Pr(>F)
1 26 44805568
2 25 11237313 1 33568255 74.68 5.594e-09 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Income should be included in the model as the model’s predictive power. The residual sum of squares decreased by ~ 33,000,000, indicating that income explains a substantial amount of variation in enrollment. The new model a significant value less than 0.001 meaning we reject the null hypothesis.
DISCLAIMER:
ChatGPT was used during the process of writing the code for the purpose of debugging and fixing errors in the code.
