A linear model can be used to describe the relationship between fuel economy(mpg) and horsepower (hp) for certain automobile models.

mpg=46.868-0.0838 hp R^2=75.5%

1) Give the symbol and the value of the correlation coefficient.

2)What would the gas mileage be for a car with a 140 horsepower engine.

3) Interpret with a sentence in context the meaning of R^2?

4) Interpret with a sentence in context the meaning of the slope.

5)Interpret with a sentence in context the meaning of the y-intercept.

6)Interpret with a sentence in context the meaning of particular car having a negative residual.

Here are a few hints to help you with some of your questions.

Coefficient of Determination is the correlation coefficient squared. To find the correlation coefficient, take the square root of .755 (75.5% converted to a decimal) for your answer. The Coefficient of Determination shows the strength of the relationship between two variables.

To find mpg, substitute 140 for hp in the equation and solve.

In the equation mpg=46.868-0.0838 hp:
46.868 represents the y-intercept and -0.0838 the slope.

I hope this will help get you started.

1) The correlation coefficient is typically denoted by the symbol "r" and it measures the strength and direction of the linear relationship between two variables. In this case, the value of the correlation coefficient is not explicitly stated, but the coefficient of determination (R^2) is given, which is closely related to r^2 and represents the percentage of variation in the dependent variable (mpg) explained by the independent variable (hp) in a linear regression model. R^2 is 75.5% in this case.

2) To find the gas mileage for a car with a 140 horsepower engine, you can substitute the value of horsepower (hp) into the linear equation: mpg = 46.868 - 0.0838 * 140. Solving this equation will give you the estimated gas mileage for a car with a 140 horsepower engine.

3) The R^2 value of 75.5% means that approximately 75.5% of the variation in gas mileage (mpg) can be explained by the variation in horsepower (hp) in the linear model. In other words, the independent variable (hp) explains 75.5% of the changes observed in the dependent variable (mpg) in this particular set of automobile models. The remaining 24.5% is attributed to other factors not accounted for in the model.

4) The slope in the linear equation represents the rate of change in gas mileage (mpg) for every unit change in horsepower (hp). In this case, the slope is -0.0838, meaning that for every one unit increase in horsepower, the gas mileage is expected to decrease by 0.0838 units. So, the slope indicates a negative relationship between horsepower and gas mileage: as horsepower increases, gas mileage tends to decrease.

5) The y-intercept in the linear equation represents the predicted value of gas mileage (mpg) when horsepower (hp) is zero or absent. In this case, the y-intercept is 46.868, which means that when the horsepower of a car is zero, the estimated gas mileage is 46.868 mpg. However, it is important to note that it may not make practical sense to interpret the y-intercept in the context of this specific problem, as having a car with zero horsepower is not feasible.

6) Residuals represent the difference between the actual observed values and the predicted values from the linear regression model. In the context of a particular car having a negative residual, it would mean that the observed gas mileage of that car is lower than the predicted value based on its horsepower according to the linear model. This suggests that there are other factors influencing the gas mileage of that specific car which are not accounted for in the linear model, resulting in a negative deviation from the predicted gas mileage.