Rework Exercise 71, coding the class marks of the history scores -2, -1, 0, 1, and 2 and the class marks of the economics scores -2, -1, 0, 1, 2, and 3. (It follows from Exercise 70 that this kind of coding will not affect the value of r.)

Exercise 71

The table at the bottom of the page shows how the history and economics scores of 25 students are distributed. Use the method of Exercise 32 to determine the value of r, replacing the row headings by the corresponding class marks (midpoints) 23, 28, 33, 38, 43, and 48 and the column headings by the corresponding class marks 23, 28, 33, 38, and 43. Use this value of r to test at the 0.05 level of significance whether there is a relationship between scores in the two subjects.

Exercise 32

In a random sample of n pairs of values of X and Y, (x_i, y_j) occurs _ij times for i = 1, 2, ... ,r and j = 1, 2, ... , c. Letting_i· denote the number of pairs where X takes on the value x_i and _·j the number of pairs where Y takes on the value y_j, write a formula for the coefficient of correlation.

Exercise 70

The calculation of r can often be simplified by adding the same constant to each x, adding the same constant to each y, or multiplying each x and/or y by the same positive constants. Recalculate r for the data of Example 7 by first multiplying each x and each y by 10 and then subtracting 70 from each x and 60 from each y

Example 7

Suppose that we want to determine on the basis of the following data whether there is a relationship between the time, in minutes, it takes a secretary to complete a certain form in the morning and in the late afternoon:

Compute and interpret the sample correlation coefficient.