/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Price, age, and horsepower In th... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 16

Price, age, and horsepower In the previous exercise, \(r^{2}=0.66\) when age is the predictor and \(R^{2}=0.69\) when both age and HP are predictors. Why do you think that the predictions of price don't improve much when HP is added to the model? (The correlation between HP and price is \(r=0.56,\) and the correlation between HP and age is \(r=-0.51 .)\)

Short Answer

Expert verified
Adding HP doesn't improve predictions much due to its correlation with age and moderate relationship with price.

Step by step solution

01

Understand the Variables

We are given three variables: price (dependent variable), age (predictor), and horsepower (HP, another predictor). Our task is to understand the impact of adding HP to the model originally predicting price using age alone.
02

Interpret R虏 and r Values

We have two R虏 values: 0.66 when predicting price using age alone, and 0.69 when predicting price using both age and HP. The correlation between HP and price is 0.56, and the correlation between HP and age is -0.51.
03

Analyze HP's Additional Predictive Power

Since the R虏 value increases from 0.66 to 0.69 with the addition of HP, the model's predictive power improves, but only slightly. The correlation between HP and price (0.56) indicates a moderate positive relationship, suggesting HP does have some predictive value, but not as strong or significant.
04

Examine Correlation Between HP and Age

The strong negative correlation of -0.51 between HP and age suggests multicollinearity or shared information between these predictors. Since HP and age are correlated, the unique information provided by HP might already be captured by age.
05

Conclusion

The improvement in prediction is small because HP adds limited unique information due to its correlation with age, and the already moderate correlation between HP and price. Thus, the data indicates that age is a stronger predictor or may overlap considerably with the information HP provides.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient
The correlation coefficient is a numerical measure that describes the strength and direction of a relationship between two variables. It is represented by the letter \( r \) and ranges from -1 to 1.
A correlation coefficient close to 1 indicates a strong positive relationship, where increases in one variable correspond with increases in the other. Conversely, a coefficient near -1 signals a strong negative relationship, meaning one variable increases as the other decreases.
  • In our exercise, the correlation between horsepower (HP) and price is 0.56, indicating a moderate positive relationship.
  • The correlation between HP and age is -0.51, which shows a moderate negative relationship.
Understanding these correlations helps determine how much one variable can explain the changes in another, which is essential when evaluating the effectiveness of predictive models.
Predictive Modeling
Predictive modeling involves creating a model that predicts an outcome based on known data. In the context of multiple linear regression, it examines the relationship between a dependent variable and multiple independent variables to predict future outcomes.
  • The exercise is an example of predictive modeling, exploring the prediction of car prices based on age and horsepower (HP).
  • The initial model, using age as the sole predictor, has an \( R^{2} \) value of 0.66, indicating that age accounts for 66% of the variability in prices.
  • Adding HP as another predictor increases the \( R^{2} \) to 0.69, suggesting that this model explains 69% of the variability. However, this increase is minimal, indicating HP adds limited unique predictive power.
Thus, successful predictive modeling depends on the quality and independence of the predictors chosen to forecast the desired outcome.
Multicollinearity
Multicollinearity occurs in multiple linear regression when two or more independent variables are highly correlated. This can cause difficulties in estimating the true effect of each predictor on the dependent variable.
  • In our exercise, the correlation of -0.51 between HP and age indicates potential multicollinearity. This suggests that these two variables might share overlapping information while predicting the price.
  • This overlap limits the additional predictive value HP can provide when included in a model with age. Hence, the small \( R^{2} \) increase from 0.66 to 0.69.
Multicollinearity can undermine the statistical significance of predictors, leading to less reliable and interpretable models. Identifying and addressing multicollinearity is crucial in enhancing the model's accuracy and reliability.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A chain restaurant that specializes in selling pizza wants to analyze how \(y=\) sales for a customer (the total amount spent by a customer on food and beverage, in pounds) depends on the location of the restaurant, which is classified as inner city, suburbia, or at an interstate exit. a. Construct indicator variables \(x_{1}\) for inner city and \(x_{2}\) for suburbia so you can include location in a regression equation for predicting the sales. b. For part a, suppose \(\hat{y}=6.9+1.2 x_{1}+0.5 x_{2} .\) Find the difference between the estimated mean sales at inner-city locations and at interstate exits.

Predicting weight Let's use multiple regression to predict total body weight (TBW, in pounds) using data from a study of female college athletes. Possible predictors are \(\mathrm{HGT}=\) height (in inches), \(\% \mathrm{BF}=\) percent body fat, and age. The display shows the correlation matrix for these variables. a. Which explanatory variable gives by itself the best predictions of weight? Explain. b. With height as the sole predictor, \(\hat{y}=-106+3.65\) (HGT) and \(r^{2}=0.55\). If you add \%BF as a predictor, you know that \(R^{2}\) will be at least \(0.55 .\) Explain why. c. When you add \% body fat to the model, \(\hat{y}=-121+\) \(3.50(\mathrm{HGT})+1.35(\% \mathrm{BF})\) and \(R^{2}=0.66 .\) When you add age to the model, \(\hat{y}=-97.7+3.43(\mathrm{HGT})+\) \(1.36(\% \mathrm{BF})-0.960(\mathrm{AGE})\) and \(R^{2}=0.67\). Once you know height and \% body fat, does age seem to help you in predicting weight? Explain, based on comparing the \(R^{2}\) values.

Suppose you fit a straight-line regression model to \(y=\) number of hours worked (excluding time spent on household chores) and \(x=\) age of the subject. Values of \(y\) in the sample tend to be quite large for young adults and for elderly people, and they tend to be lower for other people. Sketch what you would expect to observe for (a) the scatterplot of \(x\) and \(y\) and (b) a plot of the residuals against the values of age.

Cancer prediction A breast cancer study at a city hospital in New York used logistic regression to predict the probability that a female has breast cancer. One explanatory variable was \(x=\) radius of the tumor (in \(\mathrm{cm}\) ). The results are as follows: Term zf Constant -2.165 radius 2.585 The quartiles for the radius were \(\mathrm{Q} 1=1.00, \mathrm{Q} 2=1.35\), and \(Q 3=1.85\) a. Find the probability that a female has breast cancer at \(\mathrm{Q} 1\) and \(\mathrm{Q} 3 .\) b. Interpret the effect of radius by estimating how much the probability increases over the middle half of the sampled radii, between \(\mathrm{Q} 1\) and \(\mathrm{Q}_{3}\).

Hall of Fame induction Baseball's highest honor is election to the Hall of Fame. The history of the election process, however, has been filled with controversy and accusations of favoritism. Most recently, there is also the discussion about players who used performance enhancement drugs. The Hall of Fame has failed to define what the criteria for entry should be. Several statistical models have attempted to describe the probability of a player being offered entry into the Hall of Fame. How does hitting 400 or 500 home runs affect a player's chances of being enshrined? What about having a 300 average or \(1500 \mathrm{RBI} ?\) One factor, the number of home runs, is examined by using logistic regression as the probability of being elected: $$ P(\mathrm{HOF})=\frac{e^{-6.7+0.0175 \mathrm{HR}}}{1+e^{-6.7+0.0175 \mathrm{HR}}} $$ a. Compare the probability of election for two players who are 10 home runs apart - say, 369 home runs versus 359 home runs. b. Compare the probability of election for a player with 475 home runs versus the probability for a player with 465 home runs. (These happen to be the figures for Willie Stargell and Dave Winficld.)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.