Loading data

Here we load the data and ggplot then view the structure of the data.

enroll = read.csv("enrollmentForecast.csv")
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 4.0.5
str(enroll)
## '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(enroll)
##       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

Scatterplots

Here we look for any correlation between enrollment and other variables

ggplot(enroll, aes(x=ROLL, y=HGRAD)) + geom_point()

ggplot(enroll, aes(x=ROLL, y=YEAR)) + geom_point()

ggplot(enroll, aes(x=ROLL, y=INC)) + geom_point()

ggplot(enroll, aes(x=ROLL, y=UNEM)) + geom_point()

Linear model

Here we build our linear model with respect to unemployment and high school graduates to predict fall enrollment

enroll_fall = lm(ROLL ~ UNEM + HGRAD, data = enroll)
summary(enroll_fall)
## 
## Call:
## lm(formula = ROLL ~ UNEM + HGRAD, data = enroll)
## 
## 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(enroll_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

Based on F and P values, we can conclude that HGRAD is most closely related to enrollment

Residual Plot

Here we check for bias in our model

plot(enroll_fall, which=1)

plot(enroll_fall, which=4)

It appears there is some bias in our model even though the r^2 indicates our model is overall good.

Using Predict()

Now we are going to estimate the enrollment if the unemployment rate is at 9% and high school graduates are at 25,000.

enroll_predict = data.frame(UNEM = 0.09, HGRAD = 25000)
predict(enroll_fall, enroll_predict, interval="prediction")
##        fit      lwr      upr
## 1 15364.01 10461.38 20266.65

Income model

Finally, we create a model that includes income and compare to our first model using anova()

enroll_fall2 = lm(ROLL ~ UNEM + HGRAD + INC, data = enroll)
anova(enroll_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
anova(enroll_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

Based on HGRAD having a large increase to F and a decrease to P by an order of magnitudes, including the third variable improved the model.