## 34.8 Exercises

Selected answers are available in Sect. D.31.

Exercise 34.1 Draw a scatterplot with:

1. A negative correlation coefficient, with the value of $$r$$ very close to (but not equal to) $$-1$$.
2. A positive correlation coefficient, with the value of $$r$$ very close to (but not equal to) $$+1$$.
3. A correlation coefficient very close to $$0$$.

Exercise 34.2 A study (, p. 75) of American footballers measured the right-leg strengths $$x$$ of 13 players (using a weight lifting test), and the distance $$y$$ they punted a football (with their right leg) (Fig. 34.12).

1. The value of the correlation coefficient is 0.881. Compute the value of $$R^2$$, and explain what this means.
2. jamovi was used to study the correlation (Fig. 34.13). Using this output, perform a hypothesis test to determine if a correlation exists between punting distance and right-leg strength.
Exercise 34.3 A study examined the time taken to deliver soft drinks to vending machines using a sample of size $$n = 25$$ (Fig. 34.14). To perform a test of the correlation coefficient, are the statistical validity conditions met?

Exercise 34.4 A study of hot mix asphalt created $$n = 42$$ samples of asphalt and measured the volume of air voids and the bitumen content by weight (Fig. 34.15).

1. Using the plot, estimate the value of $$r$$.
2. The value of $$R^2$$ is 99.29%. What is the value of $$r$$? (Hint: Be careful!)
3. Would you expect the $$P$$-value testing $$H_0$$: $$\rho=0$$ to be small or large? Explain.
4. Would the test be statistically valid?

Exercise 34.5 The California Bearing Ratio (CBR) value is used to describe soil-sub grade for flexible pavements (such as in the design of air field runways).

One study examined the relationship between CBR and other properties of soil, including the plasticity index (PI, a measure of the plasticity of the soil).

The scatterplot from 16 different soil samples from Assam, India, is shown in Fig. 34.16.

1. Using the plot, estimate the value of $$r$$.
2. The value of $$R^2$$ is 67.07%. What is the value of $$r$$? (Hint: Be careful!)
3. Would you expect the $$P$$-value testing $$H_0$$: $$\rho=0$$ to be small or large? Explain.
4. Would the test be statistically valid?

### References

Montgomery DC, Peck EA. Introduction to regression analysis. New York: Wiley; 1992.
Myers RH. Classical and modern regression with applications. second. Duxbury; 1990.
Panda RP, Das SS, Sahoo PK. Relation between bitumen content and percentage air voids in semi dense bituminous concrete. Journal of The Institution of Engineers (India): Series A. 2018;99(2):327–32.
Talukdar DK. A study of correlation between california bearing ratio (CBR) value with other properties of soil. International Journal of Emerging Technology and Advanced Engineering. Citeseer; 2014;4(1):559–62.