Answer to Question #140750 in Statistics and Probability for Esther Gathenya

Question

c) The table below shows the hours spent by ten men on a certain task and the units of an item produced.
Men A B C D E F G H I J
Hours (X) 16 14 15 18 20 15 19 21 12 24
Units
produced(Y)

28 24 27 33 31 33 34 36 21 40

Calculate and both the Pearson correlation coefficient and coefficient of determination, interpret both and
justify in this case why it would be prudent to rely on the coefficient of determination rather than Pearson in
decision making.

Answer 1

Solution

Pearson Correlation Coefficient

$$r= {n(\sum xy)-(\sum x)(\sum y) \over \sqrt{[n \sum x^2 -{( \sum x)}^2][n \sum y^2 - {(\sum y)}^2]}}$$

$$= {10 (5512) - (174*307) \over \sqrt{[10*3148 -{(174)}^2][10(9721)-{(307)}^2]}}$$

$$=0.90142$$

Interpretation: This implies that there is a high positive relationship between the hours spent on a task by the men and the units produced.

Coefficient of determination

$$R^2= {(r)}^2$$ $$={0.90142}^2 = 0.81256$$

Interpretation: This implies that 81.256% of the variation in the units produced is due to the amount of hours spent by the men.

Justification:

It is prudent to use the coefficient of determination here because we clearly have a cause and effect relationship between the two variables, i.e. time spent (cause) affects the units of items produced (effect). There is a dependence relationship, where time spent is the independent variable and units produced the dependent variable.