Quiz on Correlation and Regression
Samantha O’Loughlin
- Q1. Describe the relationship between the two variables - bachelors level (bachelors) and per_capita_income (per_capita_per_capita_income).
- Q2. See model 1. Is the coefficient of bachelors statistically significant at 5%? Interpret the coefficient.
- Q3. See model 1. How much a typical person is predicted to make a year in a county that has 70% of its population with a bachelor’s degree?
- Q4. See model 1. What is the reported residual standard error? What does it mean?
- Q5. See model 1. What is the reported adjusted R squared? What does it mean?
- 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.
Suppose that you want to build a regression model that predicts the per_capita_income level of counties in the United States, using their bachelorsal 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 - bachelors level (bachelors) and per_capita_income (per_capita_per_capita_income).
The scatterplot is in an upward direction and the correlation is around .79, indicating that the relationship between the two variables is strong.
# 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] 0.7924464Run a regression model for per_capita_income (per_capita_per_capita_income) with one explanatory variable, bachelorsal level (bachelors), and answer Q2 through Q5.
Q2. See model 1. Is the coefficient of bachelors statistically significant at 5%? Interpret the coefficient.
Yes, it is significant because the coefficient is less than 5% at <2e-16
Q3. See model 1. How much a typical person is predicted to make a year in a county that has 70% of its population with a bachelor’s degree?
$53,255
Q4. See model 1. What is the reported residual standard error? What does it mean?
The residual standard error is l3.299. That means the model misses the actual value by $3,299
Q5. See model 1. What is the reported adjusted R squared? What does it mean?
0.6269
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 2 because it has a larger R squared.
# 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 + density, data = countyComplete)
# View summary of model 2
summary(mod_2)
##
## Call:
## lm(formula = per_capita_income ~ bachelors + density, data = countyComplete)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18257.9 -1707.7 71.5 1749.6 22101.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.319e+04 1.433e+02 92.042 < 2e-16 ***
## bachelors 4.872e+02 6.963e+00 69.973 < 2e-16 ***
## density 1.623e-01 3.499e-02 4.639 3.65e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3289 on 3140 degrees of freedom
## Multiple R-squared: 0.6305, Adjusted R-squared: 0.6303
## F-statistic: 2679 on 2 and 3140 DF, p-value: < 2.2e-16