Question

Exercise 2.143 on page 102 introduces a study examining years playing football, brain size, and percentile score on a cognitive skills test. We show computer output below for a model to predict Cognition score based on Years playing football. (The scatterplot given in Exercise 2.143 allows us to proceed without serious concerns about the conditions.) Pearson correlation of Years and Cognition \(=-0.366\) P-Value \(=0.015\) Regression Equation Cognition \(=102.3-3.34\) Years Coefficients \(\begin{array}{lrrrr}\text { Term } & \text { Coef } & \text { SE Coef } & \text { T-Value } & \text { P-Value } \\ \text { Constant } & 102.3 & 15.6 & 6.56 & 0.000 \\ \text { Years } & -3.34 & 1.31 & -2.55 & 0.015 \\ & & & & \\ & \text { S } & \text { R-sq } & \text { R-sq(adj) } & \text { R-sq(pred) } \\ 25.4993 & 13.39 \% & 11.33 \% & 5.75 \%\end{array}\) Analysis of Variance \(\begin{array}{lrrrrr}\text { Source } & \text { DF } & \text { Adj SS } & \text { Adj MS } & \text { F-Value } & \text { P-Value } \\\ \text { Regression } & 1 & 4223 & 4223.2 & 6.50 & 0.015 \\ \text { Error } & 42 & 27309 & 650.2 & & \\ \text { Total } & 43 & 31532 & & & \\ & \-- & & & \end{array}\) (a) What is the correlation between these two variables? What is the p-value for testing the correlation? (b) What is the slope of the regression line to predict cognition score based on years playing football? What is the t-statistic for testing the slope? What is the p-value for the test? (c) The ANOVA table is given for testing the effectiveness of this model. What is the F-statistic for the test? What is the p-value? (d) What do you notice about the three p-values for the three tests in parts \((\mathrm{a}),(\mathrm{b}),\) and \((\mathrm{c}) ?\) (e) In every case, at a \(5 \%\) level, what is the conclusion of the test in terms of football and cognition?

Short answer

The correlation between the years playing football and cognition score is -0.366. The p-value for testing this correlation is 0.015. The slope of the regression line is -3.34 and the t-statistic for testing the slope is -2.55. The p-value for the test is 0.015. The F-statistic for testing the effectiveness of the model is 6.50 and its p-value is 0.015. All three tests have the same p-value of 0.015. Therefore at a 5% level, we conclude that there is a statistically significant negative association between the years playing football and cognition scores.

Step-by-step solution

Get the correlation and p-value

From the Pearson correlation given, the correlation between the years and cognition is -0.366. The p-value for testing this correlation is 0.015.

Understand the regression equation

From the Regression Equation given, we can see that the slope is -3.34. This slope means that for every additional year of playing football, the cognition score decreases by 3.34 on average. The t-statistic for testing this slope is -2.55, and the p-value for the test is again 0.015. These values are found in the 'Coefficients' table.

Interpret the ANOVA table

Looking at the ANOVA table, the F-statistic for testing the effectiveness of this model is 6.50 and the p-value is 0.015.

Analyze the p-values

The p-value for the test of correlation, the test of the slope, and the test of the model from the ANOVA table are all the same and equal to 0.015.

Conclusion

Since the p-value is less than 0.05, we reject the null hypothesis in each case. This means that there is a statistically significant negative association between years playing football and cognition scores. Therefore, the evidence suggests that the more years one plays football, the lower the cognition score.