Interpreting a Computer Display
In Exercises 9–12, refer to the display obtained by using the paired data consisting of weights (pounds) and highway fuel consumption amounts (mi/gal) of the large cars included in Data Set 35 “Car Data” in Appendix B. Along with the paired weights and fuel consumption amounts, StatCrunch was also given the value of 4000 pounds to be used for predicting highway fuel consumption.





Testing for Correlation Use the information provided in the display to determine the value of the linear correlation coefficient. Is there sufficient evidence to support a claim of a linear correlation between weights of large cars and the highway fuel consumption amounts?

Verified step by step guidance

1. Step 1: Identify the linear correlation coefficient (R) from the display. The value of R is provided as -0.78762826. This coefficient measures the strength and direction of the linear relationship between the weights of large cars and their highway fuel consumption.
2. Step 2: Interpret the value of R. Since R is negative, it indicates an inverse relationship, meaning as the weight of the cars increases, the highway fuel consumption tends to decrease. The magnitude of R (close to -1) suggests a strong linear relationship.
3. Step 3: Determine the significance of the correlation. To test if the correlation is statistically significant, you would typically use a hypothesis test for the correlation coefficient. The null hypothesis states that there is no linear correlation (R = 0), while the alternative hypothesis states that there is a linear correlation (R ≠ 0).
4. Step 4: Use the sample size (n = 12) and the value of R to calculate the test statistic for the correlation coefficient. The formula for the test statistic is t = R * sqrt((n - 2) / (1 - R^2)). This test statistic can then be compared to the critical t-value from the t-distribution table with degrees of freedom df = n - 2.
5. Step 5: Based on the test statistic and the critical t-value, determine whether to reject the null hypothesis. If the test statistic exceeds the critical value, there is sufficient evidence to support the claim of a linear correlation between weights of large cars and highway fuel consumption.

Key Concepts

Linear Correlation Coefficient (R)

The linear correlation coefficient, denoted as R, measures the strength and direction of a linear relationship between two variables. Values range from -1 to 1, where -1 indicates a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 indicates no correlation. In this case, R = -0.7876 suggests a strong negative correlation between car weights and highway fuel consumption.

Prediction and Confidence Intervals

Prediction involves estimating the value of a dependent variable based on the value of an independent variable using a regression model. The confidence interval (C.I.) provides a range of values within which we expect the mean of the dependent variable to fall, while the prediction interval (P.I.) gives a range for individual future observations. The output shows a predicted fuel consumption of approximately 28.99 mpg for a car weighing 4000 pounds, with associated intervals for mean and new predictions.

Coefficient of Determination (R-squared)

R-squared (R²) quantifies the proportion of variance in the dependent variable that can be explained by the independent variable in a regression model. It ranges from 0 to 1, with higher values indicating a better fit. In this case, R² = 0.6204 suggests that about 62.04% of the variability in highway fuel consumption can be explained by the weight of the cars, indicating a moderate level of explanatory power.