Data605 Week 11

summary(cars)
##      speed           dist       
##  Min.   : 4.0   Min.   :  2.00  
##  1st Qu.:12.0   1st Qu.: 26.00  
##  Median :15.0   Median : 36.00  
##  Mean   :15.4   Mean   : 42.98  
##  3rd Qu.:19.0   3rd Qu.: 56.00  
##  Max.   :25.0   Max.   :120.00

Visulization( Plotting )

we will use plot function to plot a scatter plot with speed in x-axis and distance in the y-axis. This looks to have a linear pattern.

Creating a linear model as y(predictor) as distance and dependent variable as speed. summary function gives the intercepts = -17.5791 and x’s co-efficent as 3.9324

cars_lm <- lm(cars$dist ~ cars$speed)
cars_lm
## 
## Call:
## lm(formula = cars$dist ~ cars$speed)
## 
## Coefficients:
## (Intercept)   cars$speed  
##     -17.579        3.932
summary(cars_lm)
## 
## Call:
## lm(formula = cars$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

The equation for linear model as follows-

\[ dist = -17.579 + 3.932 * speed \]
plotting fitted regression line with actual values.

plot(cars$speed,cars$dist, main="Speed v/s distance",
xlab="speed", ylab="dist")
abline(cars_lm)

Analysis

The median value of the residuals is rougly close to zero. Quartiles and min/max values are roughly the same magnitude. P-value of speed is highly significant that proves the relation between speed and dist is strong. R-Square and Adj -Square not close to 1. So there could be another explanatory variable which is unknown that can help improve the model. note: r-square and adj r-square is used to compare different models.

Residual anaysis.

The residuals looks to be roughly scattered and dont seem have any unusual pattern.

plot(cars_lm$fitted.values, cars_lm$residuals, xlab='Fitted Values', ylab='Residuals')
abline(0,0)

qqnorm(cars_lm$residuals)
qqline(cars_lm$residuals)

The Q-Q plot of the residuals appears to slightly follow the theoretical line.Residuals are roughly normally distributed.