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.

类似问题

For the simple linear regression model, the parameter values of the model are calculated by minimize the

Question at position 38 DAEBC

Question at position 36 BCEAD

Question at position 33 CADEB

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单