Using the `mtcars` dataset in RStudio, calculate the correlation between `mpg` and `hp`, and then perform a linear regression with `mpg` as the dependent variable and `hp` as the independent variable. What is the correlation and the slope of the regression line?

Hint: Use the following code.

cor(mtcars$mpg, mtcars$hp)
lm(mtcars$mpg ~ mtcars$hp)$coefficients[2]
lm(mpg ~ hp, data=mtcars)$coefficient[2] # another method

Question

Using the `mtcars` dataset in RStudio, calculate the correlation between `mpg` and `hp`, and then perform a linear regression with `mpg` as the dependent variable and `hp` as the independent variable. What is the correlation and the slope of the regression line?

Hint: Use the following code.

cor(mtcars$mpg, mtcars$hp) 
lm(mtcars$mpg ~ mtcars$hp)$coefficients[2]
lm(mpg ~ hp, data=mtcars)$coefficient[2] # another method

BlackTom AI · Accepted Answer

We start by identifying what is being asked: compute the correlation between mpg and hp in the mtcars dataset, and then fit a simple linear regression with mpg as the dependent variable and hp as the predictor, reporting the slope of the regression line.

First, consider the correlation step: cor(mtcars$mpg, mtcars$hp).
- Correlation measures the strength and direction of a linear relationship between two variables. In the mtcars data, horsepower (hp) tends to be higher when fuel efficiency (mpg) is lower, indicating a negative relationship.
- When this computation is done on the actual dataset, the correlation is approximately -0.776, which reflects a moderately strong negative linear association.

Next, the regression step: lm(mtcars$mpg ~ mtcars$hp)$coefficients[2].
- This extracts the slope (the coefficient for hp) from the linear model predicting mpg from hp.
- The regression slope represents the expected change in mpg for a one-unit increase in hp. Given the negative relationship, we expect a negative slope.
- Using the dataset, the slope is approximately -0.068. This means that for each additional unit of horsepower, mpg is expected to decrease by about 0.068 miles per gallon, assuming a linear relationship.

An alternative method shown in the prompt, lm(mpg ~ hp, data=mtcars)$coefficient[2], yields the same slope value, just accessed via a different data specification.

Notes on interpretation and potential caveats:
- The correlation value of around -0.776 indicates a fairly strong inverse linear relationship, but it does not imply causation.
- The regression slope of -0.068 is specific to the scale of the variables in mtcars; hp is measured in horsepower and mpg in miles per gallon, so the units drive this interpretation.
- If you had transformed variables or included additional predictors, the slope and correlation could change.

If you were presented with multiple-choice options, you would look for the pair of numbers that closely matches approximately -0.776 for the correlation and -0.068 for the slope, recognizing possible minor rounding differences.

DATA1001/1901 (ND) Evaluate Quiz 5

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