pf(21186/236, df1=1, df2 = 48, lower.tail = FALSE)## [1] 1.437227e-12-Intercept t-value: -2.60
-Speed t-value: 9.46
-Speed p-value: 1.213257e-12
-48 degrees of freedom
-Multiple r^2: .6511
-FStat: 89.57 on 1 and 48 DF
-p-value: 1.49e^-12 -Speed MS: 21186
-Res MS: 236.54
-FStat: 21186/236
library(tidyverse)## ── Attaching packages ───────────────────────────────── tidyverse 1.3.0 ──## ✓ ggplot2 3.3.2 ✓ purrr 0.3.4
## ✓ tibble 3.0.3 ✓ dplyr 1.0.2
## ✓ tidyr 1.1.1 ✓ stringr 1.4.0
## ✓ readr 1.3.1 ✓ forcats 0.5.0## ── Conflicts ──────────────────────────────────── tidyverse_conflicts() ──
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()library(ISLR)
data("Carseats")
str(Carseats)## 'data.frame': 400 obs. of 11 variables:
## $ Sales : num 9.5 11.22 10.06 7.4 4.15 ...
## $ CompPrice : num 138 111 113 117 141 124 115 136 132 132 ...
## $ Income : num 73 48 35 100 64 113 105 81 110 113 ...
## $ Advertising: num 11 16 10 4 3 13 0 15 0 0 ...
## $ Population : num 276 260 269 466 340 501 45 425 108 131 ...
## $ Price : num 120 83 80 97 128 72 108 120 124 124 ...
## $ ShelveLoc : Factor w/ 3 levels "Bad","Good","Medium": 1 2 3 3 1 1 3 2 3 3 ...
## $ Age : num 42 65 59 55 38 78 71 67 76 76 ...
## $ Education : num 17 10 12 14 13 16 15 10 10 17 ...
## $ Urban : Factor w/ 2 levels "No","Yes": 2 2 2 2 2 1 2 2 1 1 ...
## $ US : Factor w/ 2 levels "No","Yes": 2 2 2 2 1 2 1 2 1 2 ...Sales and Price are numerical variables. Urban and US are categorical variables, both with two levels of yes or no.
model = lm(Sales ~ Price + Urban + US, data = Carseats)
summary(model)##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16-Price: for each additional unit of price, expected sales falls by 0.054.
-UrbanYes: if the seat is ‘Urban’ expected sales falls by about 0.022.
-USYes: if the seat is from the US, expected sales increase by 1.20.
-Equation for UrbanYes = 0 and USYes = 0: \(f(x) = 13.04 - 0.054x\)
-Equation for UrbanYes = 1 and USYes = 0: \(f(x) = 13.04 - 0.054x - 0.022\)
-Equation for UrbanYes = 0 and USYes = 1: \(f(x) = 13.04 - 0.054x + 1.2\)
-Equation for UrbanYes = 1 and USYes = 1: \(f(x) = 13.04 - 0.054x + 1.2 - 0.022\)
Based on their p-values, the null hypothesis would be rejected for the variables of Price and USYes.
model2 = lm(Sales ~ Price + US, data = Carseats)
summary(model2)##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-16anova(model)## Analysis of Variance Table
##
## Response: Sales
## Df Sum Sq Mean Sq F value Pr(>F)
## Price 1 630.03 630.03 103.0603 < 2.2e-16 ***
## Urban 1 0.10 0.10 0.0158 0.9001
## US 1 131.31 131.31 21.4802 4.86e-06 ***
## Residuals 396 2420.83 6.11
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(model2)## Analysis of Variance Table
##
## Response: Sales
## Df Sum Sq Mean Sq F value Pr(>F)
## Price 1 630.03 630.03 103.319 < 2.2e-16 ***
## US 1 131.37 131.37 21.543 4.707e-06 ***
## Residuals 397 2420.87 6.10
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Both models fit the data well. Based on the residual mean square, the model without Urban has a slightly lower residual mean square because Urban was not a statistically significant predictor of sales.
confint(model, level = .95)## 2.5 % 97.5 %
## (Intercept) 11.76359670 14.32334118
## Price -0.06476419 -0.04415351
## UrbanYes -0.55597316 0.51214085
## USYes 0.69130419 1.70984121A confidence interval of 0.95 means that we are 95% confident that the true coefficient is somewhere in the bounds of the two coefficients given in the output. This means for each additional unit of price, we are 95% confident that Sales will decrease from somewhere between 0.064 and 0.044.