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.

Finding a Prediction Interval For a car weighing 4000 pounds (x = 4000) identify the 95% prediction interval estimate of the highway fuel consumption. Write a statement interpreting that interval.

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to identify the 95% prediction interval for a car weighing 4000 pounds (x = 4000) and interpret the interval. The prediction interval provides a range within which we expect the highway fuel consumption for a new observation to fall, with 95% confidence.
2. Step 2: Locate the relevant data in the computer display. From the table, the 95% prediction interval for a car weighing 4000 pounds is given as (24.634737, 33.341609). This interval is specifically labeled as '95% P.I. for new.'
3. Step 3: Understand the difference between a prediction interval and a confidence interval. A prediction interval accounts for both the variability in the regression model and the variability of individual observations, making it wider than a confidence interval for the mean.
4. Step 4: Write the interpretation of the interval. The 95% prediction interval means that we are 95% confident that the highway fuel consumption for a car weighing 4000 pounds will fall between 24.634737 mi/gal and 33.341609 mi/gal.
5. Step 5: Note the assumptions. The prediction interval assumes that the relationship between weight and fuel consumption is linear, the residuals are normally distributed, and the sample data is representative of the population.

Key Concepts

Regression Analysis

Regression analysis is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In this context, it helps predict highway fuel consumption based on the weight of the car. The regression equation provided indicates how changes in weight affect fuel efficiency, allowing for predictions at specific weight values.

Prediction Interval

A prediction interval provides a range of values within which a new observation is expected to fall, with a certain level of confidence. In this case, the 95% prediction interval for a car weighing 4000 pounds indicates the range of highway fuel consumption values that can be expected for similar cars, accounting for variability in the data.

Correlation Coefficient

The correlation coefficient quantifies the strength and direction of a linear relationship between two variables. In this scenario, the negative correlation coefficient of -0.787 suggests a strong inverse relationship between car weight and fuel consumption, meaning that as weight increases, fuel efficiency tends to decrease. This information is crucial for interpreting the regression results.