91Ó°ÊÓ

Data for this problem are based on information from STATS Basketball Scoreboard. It is thought that basketball teams that make too many fouls in a game tend to lose the game even if they otherwise play well. Let \(x\) be the number of fouls more than (i.e., over and above) the opposing team. Let \(y\) be the percentage of times the team with the larger number of fouls wins the game. $$ \begin{array}{l|rrrr} \hline x & 0 & 2 & 5 & 6 \\ \hline y & 50 & 45 & 33 & 26 \\ \hline \end{array} $$ Complete parts (a) through (e), given \(\Sigma x=13, \quad \Sigma y=154, \quad \Sigma x^{2}=65\), \(\Sigma y^{2}=6290, \Sigma x y=411\), and \(r \approx-0.988 .\) (f) If a team had \(x=4\) fouls over and above the opposing team, what does the least-squares equation forecast for \(y\) ?

Step by step solution

01

Understand the Problem

We need to find the least-squares regression equation based on provided data, and then use it to predict the value of \( y \) when \( x = 4 \). The data given are sums \( \Sigma x, \Sigma y, \Sigma x^2, \Sigma y^2, \Sigma xy \) and the correlation coefficient \( r \).

02

Determine the Regression Equation

The regression line equation is generally of the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. The formulas to calculate the slope \( m \) and intercept \( c \) are: 1. \( m = \frac{N \Sigma(xy) - \Sigma x \Sigma y}{N \Sigma(x^2) - (\Sigma x)^2} \)2. \( c = \frac{\Sigma y - m \Sigma x}{N} \)where \( N \) is the number of data points.

03

Calculate the Slope (m)

Substitute the given values into the formula for \( m \). Here, \( N = 4 \) (because there are 4 data points). \[ m = \frac{4 \cdot 411 - 13 \cdot 154}{4 \cdot 65 - 13^2} = \frac{1644 - 2002}{260 - 169} = \frac{-358}{91} = -3.934 \]

04

Calculate the Y-Intercept (c)

Use the slope calculated in Step 3 to find the y-intercept \( c \).\[ c = \frac{154 - (-3.934) \cdot 13}{4} = \frac{154 + 51.142}{4} = \frac{205.142}{4} = 51.2855 \]

05

Form the Regression Equation

Substitute \( m \) and \( c \) into the regression line equation:\[ y = -3.934x + 51.2855 \]

06

Predict \( y \) When \( x = 4 \)

Substitute \( x = 4 \) into the regression equation:\[ y = -3.934 \times 4 + 51.2855 = -15.736 + 51.2855 = 35.5495 \]

07

Conclusion

The least-squares regression equation forecasts that if a team had 4 fouls over and above the opposing team, the percentage of times they would win is approximately 35.55%.

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Key Concepts

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

Correlation Coefficient

The correlation coefficient, often represented as \( r \), is a numerical value that helps us understand the strength and direction of a linear relationship between two variables. In this basketball context, it measures how closely the number of fouls relates to the team's winning percentage.

• A correlation coefficient value ranges from -1 to 1.
• If \( r \) is close to 1, it implies a strong positive linear relationship. This means as one variable increases, so does the other.
• If \( r \) is close to -1, it indicates a strong negative linear relationship, meaning as one variable increases, the other decreases.
• When \( r = 0 \), there's no linear relationship.

In this exercise, we have \( r \approx -0.988 \), which is a strong negative correlation. This indicates that more fouls over the opponent's result in a lower winning percentage.

Slope Calculation

The slope in a regression equation determines how steep the line is and tells us the rate at which the dependent variable \( y \) changes per unit change in the independent variable \( x \). To find the slope, you use the formula:\[ m = \frac{N \Sigma(xy) - \Sigma x \Sigma y}{N \Sigma(x^2) - (\Sigma x)^2} \] To calculate \( m \) for this example:

• We have \( N = 4 \), \( \Sigma xy = 411 \), \( \Sigma x = 13 \), \( \Sigma y = 154 \), \( \Sigma x^2 = 65 \).
• Substituting these values, we get:

\[ m = \frac{4 \cdot 411 - 13 \cdot 154}{4 \cdot 65 - 13^2} = \frac{1644 - 2002}{260 - 169} = \frac{-358}{91} \approx -3.934 \]This negative slope indicates that each additional foul over the opponent's decreases the team's winning percentage significantly.

Y-Intercept Calculation

The y-intercept \( c \) is the point at which the regression line crosses the y-axis. It tells us the predicted value of \( y \) when \( x \) is zero. In simple terms, it's the starting value of the dependent variable before any changes in the independent variable. To calculate \( c \), use:\[ c = \frac{\Sigma y - m \Sigma x}{N} \]For this problem, after finding the slope \( m = -3.934 \):

• \( \Sigma y = 154 \), \( \Sigma x = 13 \), \( m = -3.934 \), and \( N = 4 \).
• Substituting the values gives us:

\[ c = \frac{154 - (-3.934) \cdot 13}{4} = \frac{154 + 51.142}{4} = \frac{205.142}{4} \approx 51.286 \]This suggests that at zero fouls above the opponent, the team could win approximately 51.29% of their games, assuming other factors are constant.

Predictive Modeling

Predictive modeling involves using statistical techniques to predict future outcomes based on historical data. Here, the least-squares regression line serves as the predictive model to forecast basketball outcomes.Once we have the model, expressed as the equation \( y = -3.934x + 51.286 \), we can predict the team’s performance for any number of fouls.For example, if the team has 4 extra fouls over its opponent, substitute \( x = 4 \) into the equation:\[ y = -3.934 \times 4 + 51.286 = -15.736 + 51.286 = 35.55 \]

• The model predicts the team would win about 35.55% of the time.
• This prediction is useful for strategic decisions and understanding foul impacts.

Predictive modeling can be a powerful tool in sports management, helping teams understand which factors most influence their success rates.