Chapter 4, Problem 1RE

Basketball Spreads In sports betting, Las Vegas sports books establish winning margins for a team that is favored to win a game. An individual can place a wager on the game and will win if the team bet upon wins after accounting for the spread. For example, if Team A is favored by 5 points and wins the game by 7 points, then a bet on Team A is a winning bet. However, if Team A wins the game by only 3 points, then a bet on Team A is a losing bet. For NCAA Division I basketball games, a least-squares regression with explanatory variable home team Las Vegas spread, x, and response variable home team winning margin, y, is $\hat{y}=1.007x-0.012$.

Source: Justin wolfers. “Point Shaving: Corruption in NCAA Basketball”

1. a. Predict the winning margin if the home team is favored by 3 points.
2. b. Predict the winning margin (of the visiting team) if the visiting team is favored by 7 points (this is equivalent to the home team being favored by −7 points).
3. c. Interpret the slope.
4. d. Interpret the y-intercept.
5. e. The coefficient of determination is 0.39. Interpret this value.

To determine

To predict: The winning margin if the home team is favored by 3 points.

Answer to Problem 1RE

The predicted winning margin is 3.009.

Explanation of Solution

Given info:

The least square regression line relating the spread of the home team and winning point is given.

Calculation:

Substituting $x=3$ in the equation,

$\begin{array}{c}\widehat{y}=1.007x-0.012\\ =1.007\times 3-0.012\\ =3.021-0.012\\ =3.009\end{array}$

Thus, the predicted winning margin is 3.009.

To determine

To predict: The winning margin if the visiting team has 7 points.

Answer to Problem 1RE

The predicted winning margin is 3.009.

Explanation of Solution

Calculation:

The visiting team that is favored by 7 points is equivalent to the home team that is favored by –7 points. So substituting $x=-7$ in the equation,

$\begin{array}{c}\widehat{y}=1.007x-0.012\\ =1.007\times (-7)-0.012\\ =-7.049-0.012\\ =-7.061\end{array}$

Thus, the predicted winning margin of visiting team is –7.061.

To determine

To interpret: The slope parameter.

Answer to Problem 1RE

If 1 point increased in the spread, the winning margin will increase by 1.007 points.

Explanation of Solution

Interpretation:

Slope:

The coefficient value for “home team Las Vegas Spread” is 1.007.

The interpretation for slope is that, if 1 point increased in the spread, the winning margin will increase by 1.007 points.

To determine

To interpret: The y-intercept.

Answer to Problem 1RE

If the spread is by 0 points, then the home team is expected to lose by 0.012 points.

Explanation of Solution

Interpretation:

Intercept:

It is possible that home speed to have 0 spread. That is zero is reasonable for explanatory variable.

Thus, y-intercept is interpretable.

The interpretation for y-intercept is that if the spread is by 0 points, then the home team is expected to lose by 0.012 points.

To determine

To interpret: The coefficient of determination of the data.

Answer to Problem 1RE

The interpretation for the coefficient of determination is that the 39% of the variation in the winning margin is explained by the least square regression line.

Explanation of Solution

Interpretation:

The value of $R^2$ is 39%.

That is, the proportion of variability in winning margin is 39%.

Hence, it can be interpreted that the 39% of the variation in the winning margin is explained by the least square regression line.