Suppose that you want to build a regression model that predicts the income level of counties in the United States, using their educational level (Percent of population that earned a bachelor’s degree) and density (Persons per square mile). The data is obtained from countyComplete in the openintro package. To answer the questions below, replace both the name of the data and the name of the variables in the given code below.

To learn more about the countyComplete data, including the name of the variables, Google “openintro r package” and see its manual, which is usually posted as a pdf file on the CRAN website. Search countyComplete within the manual. Or click the link here.

Q1. Describe the relationship between the two variables - education level (bachelors) and income (per_capita_income).

The relationship between density per capita and bachelors degree is a strong postive correlation greater than .6. This means there must be a correlation between having a degree and income per capita.

# Load the package
library(openintro)
library(ggplot2)
str(countyComplete)
## 'data.frame':    3143 obs. of  53 variables:
##  $ state                                    : Factor w/ 51 levels "Alabama","Alaska",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ name                                     : Factor w/ 1877 levels "Abbeville County",..: 83 90 101 151 166 227 237 250 298 320 ...
##  $ FIPS                                     : num  1001 1003 1005 1007 1009 ...
##  $ pop2010                                  : num  54571 182265 27457 22915 57322 ...
##  $ pop2000                                  : num  43671 140415 29038 20826 51024 ...
##  $ age_under_5                              : num  6.6 6.1 6.2 6 6.3 6.8 6.5 6.1 5.7 5.3 ...
##  $ age_under_18                             : num  26.8 23 21.9 22.7 24.6 22.3 24.1 22.9 22.5 21.4 ...
##  $ age_over_65                              : num  12 16.8 14.2 12.7 14.7 13.5 16.7 14.3 16.7 17.9 ...
##  $ female                                   : num  51.3 51.1 46.9 46.3 50.5 45.8 53 51.8 52.2 50.4 ...
##  $ white                                    : num  78.5 85.7 48 75.8 92.6 23 54.4 74.9 58.8 92.7 ...
##  $ black                                    : num  17.7 9.4 46.9 22 1.3 70.2 43.4 20.6 38.7 4.6 ...
##  $ native                                   : num  0.4 0.7 0.4 0.3 0.5 0.2 0.3 0.5 0.2 0.5 ...
##  $ asian                                    : num  0.9 0.7 0.4 0.1 0.2 0.2 0.8 0.7 0.5 0.2 ...
##  $ pac_isl                                  : num  NA NA NA NA NA NA 0 0.1 0 0 ...
##  $ two_plus_races                           : num  1.6 1.5 0.9 0.9 1.2 0.8 0.8 1.7 1.1 1.5 ...
##  $ hispanic                                 : num  2.4 4.4 5.1 1.8 8.1 7.1 0.9 3.3 1.6 1.2 ...
##  $ white_not_hispanic                       : num  77.2 83.5 46.8 75 88.9 21.9 54.1 73.6 58.1 92.1 ...
##  $ no_move_in_one_plus_year                 : num  86.3 83 83 90.5 87.2 88.5 92.8 82.9 86.2 88.1 ...
##  $ foreign_born                             : num  2 3.6 2.8 0.7 4.7 1.1 1.1 2.5 0.9 0.5 ...
##  $ foreign_spoken_at_home                   : num  3.7 5.5 4.7 1.5 7.2 3.8 1.6 4.5 1.6 1.4 ...
##  $ hs_grad                                  : num  85.3 87.6 71.9 74.5 74.7 74.7 74.8 78.5 71.8 73.4 ...
##  $ bachelors                                : num  21.7 26.8 13.5 10 12.5 12 11 16.1 10.8 10.5 ...
##  $ veterans                                 : num  5817 20396 2327 1883 4072 ...
##  $ mean_work_travel                         : num  25.1 25.8 23.8 28.3 33.2 28.1 25.1 22.1 23.6 26.2 ...
##  $ housing_units                            : num  22135 104061 11829 8981 23887 ...
##  $ home_ownership                           : num  77.5 76.7 68 82.9 82 76.9 69 70.7 71.4 77.5 ...
##  $ housing_multi_unit                       : num  7.2 22.6 11.1 6.6 3.7 9.9 13.7 14.3 8.7 4.3 ...
##  $ median_val_owner_occupied                : num  133900 177200 88200 81200 113700 ...
##  $ households                               : num  19718 69476 9795 7441 20605 ...
##  $ persons_per_household                    : num  2.7 2.5 2.52 3.02 2.73 2.85 2.58 2.46 2.51 2.22 ...
##  $ per_capita_income                        : num  24568 26469 15875 19918 21070 ...
##  $ median_household_income                  : num  53255 50147 33219 41770 45549 ...
##  $ poverty                                  : num  10.6 12.2 25 12.6 13.4 25.3 25 19.5 20.3 17.6 ...
##  $ private_nonfarm_establishments           : num  877 4812 522 318 749 ...
##  $ private_nonfarm_employment               : num  10628 52233 7990 2927 6968 ...
##  $ percent_change_private_nonfarm_employment: num  16.6 17.4 -27 -14 -11.4 -18.5 2.1 -5.6 -45.8 5.4 ...
##  $ nonemployment_establishments             : num  2971 14175 1527 1192 3501 ...
##  $ firms                                    : num  4067 19035 1667 1385 4458 ...
##  $ black_owned_firms                        : num  15.2 2.7 NA 14.9 NA NA NA 7.2 NA NA ...
##  $ native_owned_firms                       : num  NA 0.4 NA NA NA NA NA NA NA NA ...
##  $ asian_owned_firms                        : num  1.3 1 NA NA NA NA 3.3 1.6 NA NA ...
##  $ pac_isl_owned_firms                      : num  NA NA NA NA NA NA NA NA NA NA ...
##  $ hispanic_owned_firms                     : num  0.7 1.3 NA NA NA NA NA 0.5 NA NA ...
##  $ women_owned_firms                        : num  31.7 27.3 27 NA 23.2 38.8 NA 24.7 29.3 14.5 ...
##  $ manufacturer_shipments_2007              : num  NA 1410273 NA 0 341544 ...
##  $ mercent_whole_sales_2007                 : num  NA NA NA NA NA ...
##  $ sales                                    : num  598175 2966489 188337 124707 319700 ...
##  $ sales_per_capita                         : num  12003 17166 6334 5804 5622 ...
##  $ accommodation_food_service               : num  88157 436955 NA 10757 20941 ...
##  $ building_permits                         : num  191 696 10 8 18 1 3 107 10 6 ...
##  $ fed_spending                             : num  331142 1119082 240308 163201 294114 ...
##  $ area                                     : num  594 1590 885 623 645 ...
##  $ density                                  : num  91.8 114.6 31 36.8 88.9 ...

ggplot(data = countyComplete, aes(x = bachelors , y = per_capita_income)) + #countyComplete dataset is from openintro rpackage
  geom_point()


cor(countyComplete$per_capita_income, countyComplete$bachelors, use = "pairwise.complete.obs")
## [1] 0.7924464

Run a regression model for income (per_capita_income) with one explanatory variable, educational level (bachelors), and answer Q2 through Q5.

Q2. See model 1. Is the coefficient of bachelors statistically significant at 5%? Interpret the coefficient.

Hint: One place where this information is stored is the last column on the far right, Pr (>|t|) under coefficients. One could conclude that the coefficient is significant at 0.1% if Pr < 0.001 (three stars); significant at 1% if Pr < 0.01 (two stars); and significant at 5% if Pr < 0.05 (one star). When significant, changes in the explanatory variable are highly likely to be meaningful in explaining changes in the response (or dependent) variable. The same can be said for the y-intercept. When interpreting the magnitude of the coefficient, make sure that you use the correct unit of the data. The definition of the variables can be found in the manual of the openintro package.

Yes, it is significant because its less than 4%.

Q3. See model 1. How much a typical person is predicted to make a year in a county that has

13087.68 + 494.753 * .7

Q4. See model 1. What is the reported residual standard error? What does it mean?

The residual standardis 142,091 which means the model misses the actual values by 142,090.

Q5. See model 1. What is the reported adjusted R squared? What does it mean?

The adjusted R square is .6279 which explains that there are 62.8% of varitaions in per_capita_income.

Run a second regression model for income (per_capita_income) with two explanatory variables: education (bachelors) and density (density), and answer Q6.

Q6. Compare model 1 and model 2. Which of the two models better fits the data? Discuss your answer by comparing the residual standard error and the adjusted R squared between the two models.

Model number 2 fits better with the data. Model 2 explains the variations in price per capita better. Model 2 has an adjusted R squared of .6303, which is higer than model 1 which is .6279/ Model 1 residual standard is 3299 and higher then model 2, 3289.

# Create a linear model 1
mod_1 <- lm(per_capita_income ~ bachelors, data = countyComplete)

# View summary of model 1
summary(mod_1)
## 
## Call:
## lm(formula = per_capita_income ~ bachelors, data = countyComplete)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18032.7  -1708.2     73.8   1748.0  21756.5 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13087.680    142.091   92.11   <2e-16 ***
## bachelors     494.753      6.795   72.81   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3299 on 3141 degrees of freedom
## Multiple R-squared:  0.628,  Adjusted R-squared:  0.6279 
## F-statistic:  5302 on 1 and 3141 DF,  p-value: < 2.2e-16

# Create a linear model 2
mod_2 <- lm(per_capita_income ~ bachelors, data = countyComplete)

# View summary of model 2
summary(mod_1)
## 
## Call:
## lm(formula = per_capita_income ~ bachelors, data = countyComplete)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -18032.7  -1708.2     73.8   1748.0  21756.5 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 13087.680    142.091   92.11   <2e-16 ***
## bachelors     494.753      6.795   72.81   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3299 on 3141 degrees of freedom
## Multiple R-squared:  0.628,  Adjusted R-squared:  0.6279 
## F-statistic:  5302 on 1 and 3141 DF,  p-value: < 2.2e-16