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?

Question

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?

BlackTom AI · Accepted Answer

Let's break down what the question is asking and what the provided answer implies about the relationship between mpg and hp using the correlation coefficient and the regression slope.

The prompt asks us to interpret two pieces of information from the mtcars data when examining mpg (miles per gallon) as the response and hp (horsepower) as the predictor: (1) the correlation coefficient between mpg and hp, and (2) the slope of the regression line from lm(mpg ~ hp, data=mtcars).

Option analysis:

- There is a strong negative correlation; as horsepower increases, mpg decreases.
  This statement asserts two core ideas: a negative association between mpg and hp, and a sense of strength (strong). In practice, horsepower and miles-per-gallon tend to move in opposite directions: higher horsepower generally accompanies lower fuel efficiency, so a negative correlation makes sense. Whether it is “strong” depends on the magnitude of the correlation coefficient. In the mtcars data, the correlation between mpg and hp is indeed negative and typically substantial, often in the vicinity of a moderate-to-strong negative value (for example, around -0.7 or -0.8). This aligns with the intuitive view that more powerful engines tend to be less fuel-efficient. The regression slope would also be negative, indicating that as hp increases, mpg decreases on average. Therefore, this option captures the correct directional relationship and the expected sign of both the correlation and the slope.

Notes on surrounding interpretations:
- The correlation coefficient (r) measures linear association, not causation, and ranges from -1 to 1. A negative r indicates that higher hp tends to be associated with lower mpg, but the magnitude indicates how tightly the data cluster around a straight line. A value near -0.7 or -0.8 would be considered a strong negative relationship in many contexts, though the exact categorization (weak/moderate/strong) can be subjective.
- The slope from lm(mpg ~ hp) represents the average change in mpg for a one-unit increase in hp. Because hp is measured in single horsepower units, the slope is typically a small negative number (e.g., mpg decreases by a few tenths per additional horsepower, depending on the data). The negative slope confirms the inverse relationship indicated by the correlation.
- Practical caveats: the relationship is not perfectly linear for all observations; there may be nonlinearities, heteroscedasticity, or influential points. Also, horsepower and mpg are confounded by other factors (car weight, drivetrain, engine efficiency, etc.). The correlation and slope summarize a simple bivariate relationship and do not capture all nuances of the data.

To address other potential options (if any were present):
- An option stating a positive correlation would be incorrect given the typical inverse relationship between hp and mpg.
- An option claiming no relationship or a zero slope would contradict both the observed negative slope in the regression and the negative correlation.
- An option asserting that correlation and slope show causation would be a misconception; correlation and regression describe association and prediction, not definitive causation without further evidence.

In summary, the provided statement that there is a strong negative correlation and that mpg decreases as horsepower increases correctly reflects the expected relationship between mpg and hp in the mtcars dataset, based on both the correlation coefficient and the regression slope.

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?单项选择题

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

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