/*! 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 64 For any basketball player, the n... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 64

For any basketball player, the number of points scored per game and the number of personal fouls committed are of interest. Data taken for a team last season resulted in the equation \(\hat{y}=1.122+3.394 x,\) where \(x\) is the number of personal fouls committed per game and \(y\) is the number of points scored per game. a. If one of the players committed two fouls in a game, how many points would he or she be expected to have made? b. What is the average number of points a player might expect if he or she commits three fouls in a game?

Short Answer

Expert verified
a. 7.91 points expected for two fouls in a game. b. 11.304 points expected for three fouls in a game.

Step by step solution

01

Identify values for variables

Identify the values provided by the exercise for the slope (m), the y-intercept (b) and the input variable (x). In this case we have \(m = 3.394\), \(b = 1.122\) for part a and b, \(x=2\) for part a and \(x=3\) for part b.
02

Calculate expected points scored for part a

Substitute the value of \(x =2\) into the equation \(\hat{y}=1.122+3.394 x\) to calculate the expected points scored. It gives \(\hat{y}=1.122+3.394 \cdot 2 = 1.122 + 6.788 = 7.91\)
03

Calculate expected points scored for part b

Substitute the value of \(x =3\) into the equation \(\hat{y}=1.122+3.394 x\) to calculate the expected points scored. It gives \(\hat{y}=1.122+3.394 \cdot 3 = 1.122 + 10.182 = 11.304\)

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.

Slope and Intercept in Linear Regression
In linear regression, understanding the slope and y-intercept is crucial to interpreting the equation of a line. The basic form for this equation is: \[\hat{y} = b + mx\] where:
  • \(\hat{y}\) represents the predicted value or dependent variable, such as points scored.
  • \(b\) is the y-intercept, the value of \(\hat{y}\) when \(x\) equals zero.
  • \(m\) is the slope, indicating the change in \(\hat{y}\) for every one-unit increase in \(x\), the independent variable.
In our basketball example, the equation \(\hat{y}=1.122 + 3.394x\) tells us that:
The y-intercept is 1.122. This means if no fouls are committed, the player is expected to score around 1.122 points.
The slope is 3.394, showing that for each additional foul committed, points scored increase by around 3.394.
This helps us predict outcomes (in this case, points scored) based on varying levels of fouls.
Predictive Modeling with Linear Regression
Predictive modeling is a powerful statistical tool that uses linear regression to estimate future outcomes based on past data. By using an equation derived from known data, predictions can be made for new, unseen data. This technique is particularly useful in sports analytics, business forecasting, and scientific research.
In our scenario involving basketball players, the predictive model: \[\hat{y} = 1.122 + 3.394x\] allows us to predict a player's points based on the number of fouls committed. This basketball model, driven by previous game data, assists in strategizing plays by estimating performance. For instance:
If a player commits three fouls in a game, you plug \(x = 3\) into the equation to predict the number of points scored: \(\hat{y} = 1.122 + 3.394 \times 3 = 11.304\).
This model supports coaches and players in making informed decisions about player performance and game strategies.
Mathematical Calculation in Linear Regression
The mathematical calculation aspect of linear regression involves actual computation based on the slope-intercept formula to make predictions. This is particularly important because it allows the quantification of the predictions in an easy-to-understand, numeric format.
To utilize the linear regression equation: \[\hat{y} = b + mx\] follow these steps:
  • Identify the variables: determine the slope \(m\) and the y-intercept \(b\).
  • Substitute the known value of \(x\) into the equation.
  • Perform the arithmetic operations to solve for \(\hat{y}\).
Using our example, if a player commits two fouls (\(x = 2\)), the calculation is:
\(\hat{y} = 1.122 + 3.394 \times 2 = 7.91\).
Similarly, for three fouls (\(x = 3\)), we find \(\hat{y} = 1.122 + 3.394 \times 3 = 11.304\).
These calculations highlight the practical application of mathematics in predicting sports performance through linear regression.

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.J. used linear regression to help him understand his monthly telephone bill. The line of best fit was \(\hat{y}=23.65+1.28 x,\) where \(x\) is the number of long-distance calls made during a month, and \(y\) is the total telephone cost for a month. In terms of number of long-distance calls and cost: a. Explain the meaning of the \(y\) -intercept, 23.65 b. Explain the meaning of the slope, 1.28

Explain why one would expect to find a positive correlation between the number of fire engines that respond to a fire and the amount of damage done in the fire. Does this mean that the damage would be less extensive if fewer fire engines were dispatched? Explain.

Bottled water is big business in the United States, and around the world too. Listed here are annual numbers indicating just how big the U.S. bottled water market is (volume is in gallons and producer revenues are in U.S. dollars).$$\begin{array}{lcc} \hline \text { Year } & \text { Millions of Gallons } & \text { Millions of Dollars } \\ \hline 2000 & 4,725.10 & \$ 6,113.00 \\ 2001 & 5,185.30 & \$ 6,880.60 \\ 2002 & 5,795.70 & \$ 7,901.40 \\ 2003 & 6,269,80 & \$ 8,526.40 \\ 2004 & 0,806.70 & \$ 9,169.50 \\ 2005 & 7,538.90 & \$ 10,007.40 \\ 2006 & 8,253.50 & \$ 10,857.80 \\ 2007 & 8,823.00 & \$ 11,705.90 \\ 2008 & 9,418.00 & \$ 12,573.50 \\ \hline\end{array}$$ a. Inspect the data in the chart and explain how the numbers show steady and big growth annually. b. Construct a scatter diagram using gallons, \(x,\) and dollars, \(y\) c. Does the scatter diagram show the same steady growth you discussed in part a? Explain any differences.d. Find the equation for the line of best fit. e. What does the slope found in part d represent?

Consider the saying, "Build it and they will come." This notable saying from a movie may very well apply to shopping malls. Just be sure when you build that there is room for not only the mall but also for those that will come-and thus include enough space for parking. Consider the random sample of major malls in Irvine, California.$$\begin{array}{ccc} \text { Square Feet } & \text { Parking Spaces } & \text { Number of Stores } \\\ \hline 270,987 & 3128 & 65 \\ 258,761 & 1500 & 43 \\ 1,600,350 & 8572 & 120 \\ 210,743 & 793 & 59 \\ 880,000 & 7100 & 95 \\ 2,700,000 & 15,000 & 300 \\ \hline\end{array}$$ a. Draw a scatter diagram with "parking spaces" as the dependent variable, \(y,\) and "square feet" as the independent variable, \(x .\) (Suggestion: Use 1000 s of square feet.) b. Does the scatter diagram in part a suggest that a linear regression will be useful? Explain. c. Calculate the equation for the line of best fit. d. Draw the line of best fit on the scatter diagram you obtained in part a. Explain the role of a positive slope for this pair of variables. e. Do you see a potential lurking variable? Explain its possible role. f. Draw a scatter diagram with "parking spaces" as the dependent variable, \(y,\) and "number of stores" as the independent variable, \(x\) g. Does the scatter diagram in part e suggest that a linear regression will be useful? Explain. h. Calculate the equation for the line of best fit. i. Draw the line of best fit on the scatter diagram you obtained in part e. j. Do you see a potential lurking variable? Explain its possible role. k. Draw a scatter diagram with "number of stores" as the dependent variable, \(y,\) and "square feet" as the predictor variable, \(x\) 1\. Does the scatter diagram in part \(\mathrm{k}\) suggest that a linear regression will be useful? Explain. m. Calculate the equation for the line of best fit. n. Draw the line of best fit on the scatter diagram you obtained in part \(\mathrm{k}\)

A sample of 15 upper-class students whe commute to classes was selected at registration. They were asked to estimate the distance \((x)\) and the time \((y)=\) required to commute each day to class (see the following table).$$\begin{array}{cc|cc}\begin{array}{c}\text { Distance, } x \\\\\text { (nearest mile) }\end{array} & \begin{array}{c}\text { Time, } y \\\\\text { (nearest }5 \text { minutes }) \end{array} & \begin{array}{c}\text { Distance, } x \\\\\text { (nearest mile) }\end{array} & \begin{array}{c}\text { Time, } y \\\\\text { (nearest } 5 \text { minutes } \end{array} \\\\\hline 18 & 20 & 2 & 5 \\\8 & 15 & 15 & 25 \\\20 & 25 & 16 & 30 \\\5 & 20 & 9 & 20 \\\5 & 15 & 21 & 30 \\\11 & 25 & 5 & 10 \\\9 & 20 & 15 & 20 \\\10 & 25 & & \\\\\hline \end{array}$$.a. Do you expect to find a linear relationship between the two variables commute distance and commute time? If so, explain what relationship you expect. b. Construct a scatter diagram depicting these data. c. Does the scatter diagram in part b reinforce what you expected in part a?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

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.