Using the
𝑚
𝑡
𝑐
𝑎
𝑟
𝑠
dataset in RStudio, calculate the correlation between
𝑚
𝑝
𝑔
and
ℎ
𝑝
, and then perform a linear regression with
𝑚
𝑝
𝑔
as the dependent variable and
ℎ
𝑝
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 
𝑚
𝑡
𝑐
𝑎
𝑟
𝑠
 dataset in RStudio, calculate the correlation between 
𝑚
𝑝
𝑔
 and 
ℎ
𝑝
, and then perform a linear regression with 
𝑚
𝑝
𝑔
 as the dependent variable and 
ℎ
𝑝
 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

The question asks to compute two related statistics from the mtcars dataset in R: (1) the correlation between mpg and hp, and (2) the slope of the linear regression of mpg on hp.

First, consider the provided commands conceptually. The function cor(mtcars$mpg, mtcars$hp) returns the Pearson correlation coefficient between mpg (miles per gallon) and hp (gross horsepower). In practice, across the mtcars data, mpg tends to decline as horsepower increases, yielding a negative correlation.

Next, the regression slope is obtained by fitting a model where mpg is the dependent variable and hp is the predictor, i.e., mpg ~ hp. The slope is the coefficient associated with hp in the regression output. A negative slope indicates that higher horsepower is associated with lower fuel efficiency, on average.

Now evaluate each component:

- Correlation result: The typical correlation between mpg and hp in mtcars is around -0.8, indicating a strong negative linear relationship. The exact value from the standard mtcars dataset is approximately -0.776, which aligns with the commonly reported magnitude of a strong inverse relationship. This number is unitless and falls within the range [-1, 1]. A value near -0.78 indicates that as hp increases, mpg tends to decrease in a fairly consistent linear manner, though not perfectly.

- Regression slope: When performing lm(mtcars$mpg ~ mtcars$hp), the coefficient for hp quantifies the average change in mpg per one unit increase in hp. Based on standard mtcars analyses, this slope is around -0.068, meaning that for each additional horsepower, mpg decreases by about 0.068 miles per gallon on average. This negative slope is consistent with the negative correlation, and the magnitude shows a relatively gradual decrease in mpg with increasing hp across the dataset. The slope’s sign and approximate magnitude make intuitive sense given that more powerful engines often consume more fuel.

Regarding the code snippets provided in the hint:
- cor(mtcars$mpg, mtcars$hp) should return a single numeric value close to -0.776.
- lm(mtcars$mpg ~ mtcars$hp)$coefficients[2] and lm(mpg ~ hp, data=mtcars)$coefficient[2] should both return the regression slope, which should be close to -0.068. Note that there is a slight difference in indexing or naming (coefficients vs coefficient) depending on the exact object structure returned by lm; both approaches yield the same slope if referenced correctly.

In summary:
- Correlation (mpg vs hp): approximately -0.776, indicating a strong negative linear relationship.
- Regression slope (mpg ~ hp): approximately -0.068, indicating mpg decreases by about 0.068 for each additional horsepower on average.

If you had multiple-choice options, you would select the option that reports a correlation near -0.776 and a slope near -0.068. Since no answer options were provided in the prompt, this explanation connects the expected numerical results to the methods described by the hint.

类似问题

A researcher collected two datasets. In Dataset A, the correlation between two measures was .60. In Dataset B, the correlation between two different measures was .30. Using this information, what can we conclude about the regression slopes in each dataset?

位置1的问题 Consider two variables, X and Y. The sign of the correlation coefficient describing the linear relationsip between X and Y has the same sign as the corresponding simple linear regression coefficient.TrueFalseDon't Know清除选择

Which of the following statements is true?

In RStudio, run the following R code on the `mtcars` dataset: > cor(mtcars$mpg, mtcars$hp) > lm(mpg ~ hp, data=mtcars) What does the correlation coefficient and the slope of the regression line tell you about the relationship between mpg and hp?

Which one of the following statements is true?

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