Homework exercise

To be solved at home before the exercise session.

    1. Show that if in simple linear regression both the explanatory variable \(x\) and the response \(y\) have been marginally standardized such that \(\bar{x} = 0, s_x = 1\) and \(\bar{y} = 0, s_y = 1\), then the estimated least squares regression model is simply, \[ \hat{y}_i = \hat{\rho}(x, y) x_i. \] That is, the regression coefficient of \(x\) equals the sample correlation between \(x\) and \(y\).

    2. The cars data give the speeds of cars (speed, in mph) and the corresponding distances taken to stop (dist, in feet). The below shows the model summary of a simple linear regression model fit using speed as an explanatory variable and dist as a response. Interpret the model results.

cars_lm <- lm(dist ~ speed, data = cars)
summary(cars_lm)
## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## 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 *  
## 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
plot(dist ~ speed, data = cars)
abline(cars_lm)

Class exercise

To be solved at the exercise session.

  1. The file data_children.txt contains data on children’s ages (age, in months) and heights (height, in centimeters). Investigate whether there is a linear relationship between the two variables.
    1. Read the file into R using the command read.table.
    2. Draw a scatter plot of age and height.
    3. Fit a linear model to the data using height as a response variable.
    4. Add the fitted regression line to the scatter plot. Does the fit appear good?
    5. Interpret the estimated regression coefficient of age and the \(R^2\)-value of the model.
  1. The file data_tobacco.txt contains data on cigarette consumption and lung cancer incidences from 11 different countries. The variable consumption describes the yearly consumption of cigarettes per capita in 1930 and the variable incidence tells the lung cancer incidence rates per 100 000 people in 1950. (Recall exercise 7.2)
    1. Read the file into R using the command read.table.
    2. Draw a scatter plot of consumption and incidence.
    3. Fit a linear model to the data using incidence as a response variable.
    4. Add the fitted regression line to the scatter plot. Does the fit appear good?
    5. Interpret the estimated regression coefficient and \(p\)-value of consumption.
    6. Interpret the \(R^2\)-value of the model.
    7. Drop USA from the data, redo the previous analysis and compare the results to those obtained with the full data. What happened?
  1. (Optional) Investigate how much a single outlier can affect the results of a linear model: Create a small data set that has a perfect linear relationship between its two variables (such a model has the explanatory variable \(p\)-value equal to 0 and the coefficient of determination equal to 1). Then, add a single outlying data point and see how much you can change the \(p\)-value and the coefficient of determination by varying the outlier’s value.