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1 Question 1

A researcher was interested in the relationship between the age of drivers and the maximum distance at which a newly designed sign is legible. Data was collected from a random sample of 30 drivers. The age of the driver and the maximum distance at which they could read the newly designed road sign were recorded.

The data is stored in the file road.csv and contains the following variables for each driver:

Variable	Description
Distance	the maximum distance at which the driver could read the sign (in metres).
Age	the age of the driver (in years).

Make sure you change your name and UPI/ID number at the top of the assignment.
Comment on the scatter plot of the data.
A simple linear regression model has been fitted and output provided. Stick with the simple linear regression model. (DO NOT CHANGE THE FITTED MODEL.)
Create a scatter plot with the fitted line from the fitted model superimposed over it.
Complete the equation of the fitted model as part of the Method and Assumption Checks.
Complete the “% of the variability” statement as part of the Method and Assumption Checks.
Complete the Executive Summary by adding two more sentences. One sentence should be interpreting the relevant strength of evidence in context and the other should be estimating, in context, the effect of a 10 year increase in the age of the driver.

1.1 Question of interest/goal of the study

We are interested in whether the age of the driver affects the maximal distance that a road sign is legible.

1.2 Read in and inspect the data:

Road.df=read.csv("road.csv",header=T)
plot(Distance~Age, main="Maximum legible distance versus Age",data=Road.df)

road.fit = lm(Distance ~ Age, data=Road.df)
summary(road.fit)

## 
## Call:
## lm(formula = Distance ~ Age, data = Road.df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.876 -12.726   2.327  10.253  33.173 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 175.7648     7.1561  24.562  < 2e-16 ***
## Age          -0.9165     0.1294  -7.084 1.05e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.17 on 28 degrees of freedom
## Multiple R-squared:  0.6419, Adjusted R-squared:  0.6291 
## F-statistic: 50.18 on 1 and 28 DF,  p-value: 1.046e-07

1.3 Comment on the plots

There looks to be a negative relationship between age and maximum legible distance; as age increases, the maximum legible distance decreases.

1.4 Fit an appropriate linear model, including model checks and relevant output. DO NOT CHANGE THE FITTED MODEL

Road.lm=lm(Distance~Age,data=Road.df)
modelcheck(Road.lm)

summary(Road.lm)

## 
## Call:
## lm(formula = Distance ~ Age, data = Road.df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -23.876 -12.726   2.327  10.253  33.173 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 175.7648     7.1561  24.562  < 2e-16 ***
## Age          -0.9165     0.1294  -7.084 1.05e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 15.17 on 28 degrees of freedom
## Multiple R-squared:  0.6419, Adjusted R-squared:  0.6291 
## F-statistic: 50.18 on 1 and 28 DF,  p-value: 1.046e-07

confint(Road.lm)

##                  2.5 %      97.5 %
## (Intercept) 161.106186 190.4234214
## Age          -1.181517  -0.6514817

1.5 Create a scatter plot with the fitted line from your model superimposed over it.

plot(Distance~Age, main="Maximum legible distance versus Age",data=Road.df)

plot(Distance~Age,data=Road.df, xlim=c(0,80), ylim=c(0,180))
abline(road.fit, lty=2, col = "green")
prediction=predict(road.fit, newdata=data.frame(Age=c(20,40,60,80)))
points(c(20,40,60,80), prediction, col = "green", pch=19)

1.6 Method and Assumption Checks

As the initial plot of the data shows a linear relationship, we have fitted a linear regression model to our data. The data is from a random sample so we can assume independence. The residual plot shows randomness around zero with constant variance. The residuals look like they come from a normal distribution. There are no overly unduly influential points. All assumptions appear to be satisfied. The effect of the driver’s age is statistically significant.

1.6.1 Complete the equation below:

Our model is:

{Distance = 175.7648 - 0.9165 x Age + \(\epsilon\)} where \(\epsilon_i \sim iid ~ N(0,\sigma^2)\)

1.6.2 Complete the statement

Our model explains 62.91% of the variation in the response variable.

1.7 Executive Summary