Dan Wigodsky
Assignment 11
April 22, 2018
Simple Linear Regression
……………………………………..
Does the stopping distance for a car have a relationship to its speed? Can its speed be used to predict its stopping distance? We perform a linear regression to find if there is a relationship.
model.for.stopping.distance<-lm(cars$dist~cars$speed)
summary(model.for.stopping.distance)
##
## Call:
## lm(formula = cars$dist ~ cars$speed)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29.069 -9.525 -2.272 9.215 43.201
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.5791 6.7584 -2.601 0.0123 *
## cars$speed 3.9324 0.4155 9.464 1.49e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared: 0.6511, Adjusted R-squared: 0.6438
## F-statistic: 89.57 on 1 and 48 DF, p-value: 1.49e-12
Our model is statistically significant at the .001 significance level. Our standard error for our slope is reasonably smaller than the slope coefficient. To validate our model, we need to make sure our relationship is appropriately linear and variance- stable. There appears to be a quadratic relationship, but is the linear model statistically inappropriate? To find this out, we explore our residuals.
Our residuals are reasonably normal. We can see that in the quartiles in our summary and in the plots below. At 1.49* 10-12, our p-value is incredibly small and is significant at the .001 significance level. Our R2 is .6511 and the adjusted R2 is close to that. This means most of the difference in stopping distance can be explained by our model.
## [1] "The mean of the residuals is "
## [1] 8.65974e-17
##
## studentized Breusch-Pagan test
##
## data: model.for.stopping.distance
## BP = 3.2149, df = 1, p-value = 0.07297
Our residuals are nearly normal and stable and we accept our model: stopping distance = -17.5791 + 3.9324 * speed. According to the cars data set, there is a relationship between speed and stopping distance with stopping distance as the dependent variable.
April 22, 2018
Simple Linear Regression
……………………………………..
Does the stopping distance for a car have a relationship to its speed? Can its speed be used to predict its stopping distance? We perform a linear regression to find if there is a relationship.
model.for.stopping.distance<-lm(cars$dist~cars$speed)
summary(model.for.stopping.distance)##
## Call:
## lm(formula = cars$dist ~ cars$speed)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29.069 -9.525 -2.272 9.215 43.201
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -17.5791 6.7584 -2.601 0.0123 *
## cars$speed 3.9324 0.4155 9.464 1.49e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.38 on 48 degrees of freedom
## Multiple R-squared: 0.6511, Adjusted R-squared: 0.6438
## F-statistic: 89.57 on 1 and 48 DF, p-value: 1.49e-12Our model is statistically significant at the .001 significance level. Our standard error for our slope is reasonably smaller than the slope coefficient. To validate our model, we need to make sure our relationship is appropriately linear and variance- stable. There appears to be a quadratic relationship, but is the linear model statistically inappropriate? To find this out, we explore our residuals.
Our residuals are reasonably normal. We can see that in the quartiles in our summary and in the plots below. At 1.49* 10-12, our p-value is incredibly small and is significant at the .001 significance level. Our R2 is .6511 and the adjusted R2 is close to that. This means most of the difference in stopping distance can be explained by our model.
## [1] "The mean of the residuals is "## [1] 8.65974e-17##
## studentized Breusch-Pagan test
##
## data: model.for.stopping.distance
## BP = 3.2149, df = 1, p-value = 0.07297Our residuals are nearly normal and stable and we accept our model: stopping distance = -17.5791 + 3.9324 * speed. According to the cars data set, there is a relationship between speed and stopping distance with stopping distance as the dependent variable.