Ch. 10 - Correlation and Regression

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 10 - Correlation and Regression

Problem 10.3.15

Chapter 10, Problem 10.3.15

Finding a Prediction Interval
In Exercises 13–16, use the following paired data consisting of weights of large cars (pounds) and highway fuel consumption (mi/gal) from Data Set 35 “Car Data” in Appendix B. (These are the same data used in Exercises 9-12.) Let x represent the weight of the car and let y represent the corresponding highway fuel consumption. Use the given weight and the given confidence level to construct a prediction interval estimate of highway fuel consumption.

Cars Use x = 3800 pounds with a 99% confidence level.

Verified step by step guidance

Step 1: Organize the data. The table provides paired data for car weights (x) and highway fuel consumption (y). Identify the given weight (x = 3800 pounds) and note the confidence level (99%).

Step 2: Perform a linear regression analysis. Use the paired data to calculate the regression equation, which is typically in the form y = b0 + b1*x, where b0 is the y-intercept and b1 is the slope. This requires calculating the mean and standard deviation of x and y, as well as the correlation coefficient (r).

Step 3: Calculate the standard error of the estimate (SE). The formula for SE is SE = sqrt(Σ(y - y_pred)^2 / (n - 2)), where y_pred is the predicted value from the regression equation, and n is the number of data points.

Step 4: Determine the prediction interval. Use the formula y_pred ± t*SE*sqrt(1 + 1/n + (x - x_mean)^2 / Σ(x - x_mean)^2), where t is the critical value from the t-distribution corresponding to the 99% confidence level and degrees of freedom (df = n - 2).

Step 5: Interpret the prediction interval. The result will provide a range within which the highway fuel consumption is expected to fall for a car weighing 3800 pounds, with 99% confidence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Prediction Interval

A prediction interval provides a range of values within which a future observation is expected to fall, given a certain level of confidence. It is calculated using the mean of the predicted values, the standard error of the estimate, and a critical value from the t-distribution based on the desired confidence level. This interval accounts for both the variability in the data and the uncertainty in the prediction.

Confidence Level

The confidence level indicates the degree of certainty that the prediction interval will contain the true value of the dependent variable. Common confidence levels are 90%, 95%, and 99%. A higher confidence level results in a wider prediction interval, reflecting greater uncertainty about the estimate, while a lower confidence level yields a narrower interval.

Linear Regression

Linear regression is a statistical method used to model the relationship between a dependent variable and one or more independent variables. In this context, the weight of the car (independent variable) is used to predict highway fuel consumption (dependent variable). The regression equation is derived from the data, allowing for predictions and the calculation of prediction intervals based on the fitted model.

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Textbook Question

Testing for a Linear Correlation

In Exercises 13–28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of α = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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.

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