/*! 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 34 In Exercise 9.26 on page 538 we ... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 34

In Exercise 9.26 on page 538 we consider simple linear models to predict winning percentages for NBA teams based on either their offensive ability (PtsFor = average points scored per game) or defensive ability (PtsAgainst \(=\) average points allowed per game). With multiple regression we can include both factors in the same model. Use the data in NBAStandings to fit a two-predictor model for WinPct based on PtsFor and PtsAgainst. (a) Write down the fitted prediction equation. (b) The Dallas Mavericks (2011 NBA Champion) had a 0.695 winning percentage over the season, scoring 100.2 points per game, while allowing 96.0 points against per game. Find the predicted winning percentage for the Mavericks using this model and compute the residual. (c) Comment on the effectiveness of each predictor in this model. (d) As a single predictor, PtsAgainst is more effective than PtsFor. Do we do much better by including both predictors? Choose some measure (such as \(s_{\epsilon}, \operatorname{SSE},\) or \(\left.R^{2}\right)\) to compare the simple linear model based on PtsAgainst to this two-predictor model.

Short Answer

Expert verified
First, we create a prediction equation based on our two predictors, PtsFor and PtsAgainst. Then, we use the Mavericks' scores to predict their WinPct and compute the residual. Each predictor's effectiveness is then analyzed by checking their coefficient values. Finally, we compare the two-predictor model with the simple linear model by comparing their \(R^2\), where a significantly higher \(R^2\) in the two-predictor model would indicate a better fit. Exact outcomes will depend on the given data.

Step by step solution

01

Writing the Fitted Prediction Equation

The general form of the multiple regression equation is:\(Y = b_0 + b_1X_1 + b_2X_2 + e\)In this case, 'Y' is WinPct, 'X_1' is PtsFor, and 'X_2' is PtsAgainst. The coefficients (b_0, b_1, b_2) are yet to be determined based on the data in NBAStandings.
02

Calculating the Predicted Winning Percentage for Dallas Mavericks

Once the coefficients are determined from the data, we apply the score of the Dallas Mavericks in the above equation. Calculating the predicted WinPct value for the Mavericks using the formula from Step 1 and given stats (PtsFor = 100.2, PtsAgainst = 96.0). Then, we find the residual by subtracting the observed value (WinPct = 0.695) from the predicted value.
03

Commenting on the Effectiveness of Each Predictor

We analyze the coefficient of each predictor in the model. The absolute value of each coefficient indicates its effectiveness. A higher absolute value means more influence of the predictor on the outcome.
04

Comparing Models

To compare the model using both predictors (PtsFor and PtsAgainst) to the simple linear model based on PtsAgainst, we can use the value of \(R^2\) for the two models. If \(R^2\) is significantly higher for the two-predictor model, that means including both predictors improves the predictive ability over using just PtsAgainst.

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Key Concepts

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

Linear Regression Models
Linear regression models are a statistical tool used to understand the relationship between one or more independent variables and a dependent variable. In the context of predicting NBA teams' winning percentages, a multiple linear regression model can include independent variables like the average points scored per game (PtsFor) and the average points allowed per game (PtsAgainst).

The main goal here is to see how these variables affect the winning percentage or to predict the winning percentage based on these inputs. In a simple linear regression model, you have one predictor and one response variable, allowing for an easy-to-understand relationship between the two.
  • In a simple model: the formula is of the form \( Y = b_0 + b_1X + e \).
  • In multiple regression: more than one predictor is included, thus \( Y = b_0 + b_1X_1 + b_2X_2 + e \).
Each added variable allows for a more nuanced prediction, as each predictor can independently contribute to the outcome variable, showing its unique potency in affecting the result.
Prediction Equation
A prediction equation in the context of regression comprises the formula developed based on data to estimate the dependent variable outcomes using the predictors. From our multiple regression exercise, the prediction equation would involve determining coefficients for our predictors (PtsFor and PtsAgainst) to predict the winning percentage (WinPct) of an NBA team.

The generalized form of this prediction equation is: \[ Y = b_0 + b_1X_1 + b_2X_2 + e \] This specific equation allows one to plug in specific game statistics (PtsFor and PtsAgainst) to determine a team's likely performance (WinPct) in terms of their winning percentage. This formula must be fitted using real data, where coefficients \( b_0 \), \( b_1 \), and \( b_2 \) are determined through statistical methods like the least squares approach.
  • \( b_0 \): The intercept, representing the predicted WinPct when both PtsFor and PtsAgainst are zero.
  • \( b_1 \): The coefficient for PtsFor, showing the predicted change in WinPct for each additional point scored.
  • \( b_2 \): The coefficient for PtsAgainst, showing the predicted change in WinPct for each additional point allowed.
Regression Coefficients
Regression coefficients are values that multiply the predictor variables in a regression equation. They provide crucial insights into the weight each predictor variable holds in predicting an outcome.
In our NBA example, after fitting the model, you'd obtain coefficients \( b_1 \) and \( b_2 \) for PtsFor and PtsAgainst, respectively. These numbers tell you how much a team鈥檚 predicted winning percentage changes with a one-unit increase in either PtsFor or PtsAgainst. Understanding these coefficients helps determine which variables are most influential.
  • If \( |b_1| > |b_2| \), PtsFor has a greater impact on predicting WinPct than PtsAgainst and vice versa.
  • A coefficient with a larger absolute value signifies a more significant relationship between the independent variable and the dependent variable.
Hence, examining these coefficients helps in evaluating which strategy鈥攊mproving offensive or defensive ability鈥攚ould most effectively enhance team performance.

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Most popular questions from this chapter

Use information in the ANOVA table below, which comes from fitting a multiple regression model to predict the prices for horses (in \(\$ 1000 \mathrm{~s}\) ). $$\begin{array}{lrrrrr} \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\\\text { Regression } & 3 & 4327.7 & 1442.6 & 10.94 & 0.000 \\ \text { Residual Error } & 43 & 5671.4 & 131.9 & & \\\\\text { Total } & 46 & 9999.1 & & &\end{array}$$ How many predictors are in the model?

In Exercises 9.25 and 9.64 we attempt to predict a country's life expectancy based on the percent of government expenditure on health care, using a sample of fifty countries in the dataset SampCountries. We now add to the model the variables population (in millions), percentage with Internet, and birth rate (births per 1000 ). Table 10.1 shows some computer output from fitting this model. (a) Which variables are significant predictors of life expectancy in this model, at a \(5 \%\) level? Which is the most significant? (b) Predict the life expectancy of a country that spends \(20 \%\) of government expenditures on health care, has a population of \(2,500,000,\) for which \(75 \%\) of people have access to the Internet, and the birth rate is 30 births per 1000 . (c) How does predicted life expectancy change if we find out the Internet rate is actually higher? $$\begin{aligned}&\text { Table 10.1 } \text { Multiple regression output }\\\ &\begin{array}{ccccc} \hline & \text { Estimate } & \text { Std. Error } & \text { t-value } & \operatorname{Pr}(>|t|) \\ \hline \text { (Intercept) } & 77.30 & 4.11 & 18.788 & <2 \mathrm{e}-16 \\ \text { Health } & 0.133 & 0.216 & 70.613 & 0.543 \\ \text { Population } & -0.017 & 0.029 & -0.573 & 0.570 \\ \text { Internet } & 0.112 & 0.047 & 2.347 & 0.023 \\ \text { BirthRate } & -0.555 & 0.115 & -4.824 & <0.0001 \\\\\hline\end{array}\end{aligned}$$

Use information in the ANOVA table below, which comes from fitting a multiple regression model to predict the prices for horses (in \(\$ 1000 \mathrm{~s}\) ). $$\begin{array}{lrrrrr} \text { Source } & \text { DF } & \text { SS } & \text { MS } & \text { F } & \text { P } \\\\\text { Regression } & 3 & 4327.7 & 1442.6 & 10.94 & 0.000 \\ \text { Residual Error } & 43 & 5671.4 & 131.9 & & \\\\\text { Total } & 46 & 9999.1 & & &\end{array}$$ How many horses are in the sample?

Using the data in StudentSurvey, we see that the regression line to predict Weight from Height is \widehat{Weight } \(=-170+4.82\) Height. Figure 10.8 shows three graphs for this linear model: the scatterplot with least squares line, a histogram of the residuals, and a scatterplot of residuals against predicted values. (a) One of the students in the dataset has a height of 63 inches and a weight of 200 pounds. Put an arrow showing the dot representing this person on the scatterplot with least squares line (or a rough sketch of the plot). (b) Calculate the predicted value and the residual for the person described in part (a). (c) Put an arrow showing where the person from part (a) is represented in the histogram of residuals. Also, put an arrow showing where the person from part (a) is represented in the scatterplot of residuals against predicted values. (d) Determine whether the conditions are met for inference on this regression model.

Predicting Length of Games in Baseball Baseball is played at a fairly leisurely pace-in fact, sometimes too slow for some sports fans. What contributes to the length of a major league baseball game? The file BaseballTimes contains information from a sample of 30 games to help build a model for the time of a game (in minutes). Potential predictors include: Runs \(\quad\) Total runs scored by both teams Margin Difference between the winner's and loser's scores Hits Total base hits for both teams Errors Total number of errors charged to both teams Pitchers Total number of pitchers used by both 1 teams Walks Total number of walks issued by pitchers from both teams (a) Use technology to find the correlation between each of the predictors and the response variable Time. Identify the predictors that appear to be potentially useful based on these correlations. (b) Try different models and combinations of predictors to help explain the game times. Try to get a good \(R^{2}\) and a good ANOVA p-value, but also have significant predictors. Decide on a final model and briefly indicate why you chose it.
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