Question

Exercise 2.143 on page 102 introduces a study that examines several variables on collegiate football players, including the variable Years, which is number of years playing football, and the variable Cognition, which gives percentile on a cognitive reaction test. Exercise 2.182 shows a scatterplot for these two variables and gives the correlation as -0.366 . The regression line for predicting Cognition from Years is: $$\text { Cognition }=102-3.34 \cdot \text { Years }$$ (a) Predict the cognitive percentile for someone who has played football for 8 years and for someone who has played football for 14 years. (b) Interpret the slope in terms of football and \(\operatorname{cog}-\) nitive percentile. (c) All the participants had played between 7 and 18 years of football. Is it reasonable to interpret the intercept in context? Why or why not?

Short answer

a) When playing football for 8 years, the cognitive percentile is approximately 75.28, and for 14 years, it is approximately 58.24. b) The slope of -3.34 indicates that for each additional year playing football, there is an expected decrease of 3.34 percentile in cognitive reaction test score. c) It is not reasonable to interpret the intercept in this context because it represents the cognitive percentile when Years is zero, which falls outside the given years range of 7 to 18.

Step-by-step solution

Predict cognitive percentile based on years playing football

Substitute the given years values into the regression equation, that is, for 8 years, calculate \(Cognition = 102 - 3.34 * 8\), and for 14 years, calculate \(Cognition = 102 - 3.34 * 14.\)

Interpret the slope in context

The slope of the regression line is -3.34. It represents the change in the dependent variable (Cognition) for each unit increase in the independent variable (Years). In the context of the problem, the slope can be interpreted as the expected decrease in cognitive percentile for each additional year playing football.

Interpret the intercept in context

The intercept is the value of the dependent variable (Cognition) when the independent variable (Years) is zero. However, since the range of Years all participants had played football is between 7 and 18, it is not reasonable to interpret the intercept in this context, as it falls outside the scope of the data set.

Key concepts

Cognitive Percentile Prediction

Understanding cognitive percentile prediction through regression analysis is pivotal in many statistical applications. In the context of the textbook example, the ability to predict an individual's cognitive test percentile based on the number of years they've played football is assessed through a regression equation. To do this, one substitutes the number of years played into the given regression equation. For example, to predict the cognitive percentile for a player with 8 years of football, one would compute:
\[ Cognition = 102 - 3.34 \times 8. \]
For 14 years, the equation would be
\[ Cognition = 102 - 3.34 \times 14. \]
These calculations yield the expected cognitive percentile for the respective years played. It's important to note that this prediction assumes a linear relationship between years of playing football and cognition, as implied by the data and the correlation provided in the study.

Regression Equation Interpretation

The interpretation of a regression equation is fundamental in making sense of the relationship between variables. In our example, the equation
\[ Cognition = 102 - 3.34 \times Years \]
indicates that the starting point for cognitive percentile (when no years are played) is 102. However, as the number of years increases, the cognitive percentile decreases. The negative sign in front of the 3.34 coefficient tells us that there is an inverse relationship between playing football and cognitive percentile, according to the data. This means that as the Years variable increases, the Cognition score tends to decrease. Understanding this relationship allows researchers and students alike to draw inferences and make predictive statements about the data set in question.

Slope and Intercept Analysis

In regression analysis, slope and intercept are the cornerstone to understanding the dynamics of the relationship between variables. The slope, -3.34, in our equation represents the average change in cognitive percentile for each additional year of playing football.
It indicates that, on average, a player's cognitive percentile is expected to decrease by 3.34 points for each year they continue to play football. This factor is crucial for understanding the potential impact of playing time on cognition as seen in the study.
The intercept, 102, theoretically represents the cognitive percentile when a player has zero years of football experience. Since the range of Years in the study is from 7 to 18, the intercept can't be applied directly to the context because it represents an extrapolation beyond the observed data range. Interpreting the intercept requires caution since it may not hold true for values outside the studied interval, such as someone who has never played football.