12.6 -Simple Linear Regression Example

12.6 -Simple Linear Regression Example

Example: Vertical jump to predict 40-yd dash time

Each spring in Indianapolis, IN the National Football League (NFL) holds its annual combine. The combine is a place for college football players to perform various athletic and psycholological tests in front of NFL scouts.

Two of the most recognized tests are the forty-yard dash and the vertical jump. The forty-yard dash is the most popular test of speed, while the vertical jump measures the lower body power of the athlete.

Football players train extensively to improve these tests. We want to determine if the two tests are correlated in some way. If an athlete is good at one will they be good at the other? In this particular example, we are going to determine if the vertical jump could be used to predict the forty-yard dash time in college football players at the NFL combine.

Data from the past few years were collected from a sample of 337 college football players who performed both the forty-yard dash and the vertical jump.


Solution

We can learn more about the relationship between these two variables using correlation and regression.

Correlation
Is there an association?

The correlation is given in the Minitab Express output:

Pearson correlation for Forty and Vertical = -0.729589

P-Value = <0.0001

The variables have a strong, negative association.

Simple Linear Regression
Can we predict 40-yd dash time from vertical jump height?

Next, we'll conduct the simple linear regression procedure to determine if our explanatory variable (vertical jump height) can be used to predict the response variable (40 yd time).

Assumptions

First, we'll check our assumptions to use simple linear regression.

Assumption 1: Linearity

The scatterplot below shows a linear, moderately negative relationship between the vertical jump (in inches) and the forty-yard dash time (in seconds).

Scatterplot of forty yard dash time vs vertical jump height

Assumption 2: Independence of Errors

The correlation shown in the Versus Fits scatterplot is approximately 0. This assumption is met.

Versus fits plot for vertical jump vs forty-yard dash time


Assumption 3: Normality of Errors

The normal probability plot and the histogram of the residuals confirm that the distribution of residuals is approximately normal.

Normall probability plot of residuals for forty yard dash and vertical height

Histogram of residuals of forty yard time vs vertical jump height

Assumption 4: Equal Variances

Again, using the Versus Fits scatterplot we see no pattern among the residuals. We can assume equal variances.

Versus fits scatterplot for vertical jump height vs forty yard dash time

 

ANOVA Table

The ANOVA source table gives us information about the entire model. The \(p\) value for the model is <0.0001. Because this is simple linear regression (SLR), this is the same \(p\) value that we found earlier when we examined the correlation and the same \(p\) value that we see below in the test of the statistical significance for the slope. Our \(R^2\) value is 0.5323 which tells us that 53.23% of the variance in forty-yard dash times can be explained by the vertical jump height of the athlete. This is a fairly good \(R^2\) for SLR.

Analysis of Variance
Source DF Adj SS Adj MS F-Value P-Value
Regression 1 15.8872 15.8872 381.27 <0.0001
Error 335 13.9591 0.0417    
Total 336 29.8464      
Model Summary
S R-sq R-sq(adj)
0.204130 53.23% 53.09%
Coefficients
Predictor Coef SE Coef T-Value P-Value
Constant 6.50316 0.09049 71.87 <0.0001
Vertical -0.053996 0.002765 -19.53 <0.0001
Regression Equation
The regression equation for the two variables is:

Forty = 6.50316 - 0.053996 Vertical

The regression equation indicates that for every inch gain in vertical jump height the 40-yd dash time will decrease by 0.053996 (the slope of the regression line). Finally, the fitted line plot shows us the regression equation on the scatter plot.

Scatterplot of vertical jump vs forty-yard dash times

View the video for this example showing the steps in Minitab Express.


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