STATISTIC AND PROBABILITY

Question

In a study of the factors that affect success in economics, data were collected for 8 business

students. Scores on a calculus placement test are given with economics final exam scores. The

data are below. Calculate the correlation coefficient between the two variables. Show the complete

calculation and interpret the coefficient value.

CALCULUS PLACEMENT SCORE	EXAM FINAL SCORE
17	73
21	66
11	64
16	61
15	70
11	71
24	90
27	68

Answer 1

Easiest way to get this solution is to use Excel or a similar software. Will show both ways to illustrate how the correlation coefficient is derived. Using the following formula, with n = 8.

r = [ n(Σxy) - ΣxΣy ] / square root ( [nΣx² - (Σx²)] [nΣy² - (yx²)] )

With the summations in Excel, we have

results in r = .3909

Alternatively, you can use Excel CORREL function, select all x's for one array and all y's for second array, that is =CORREL(A2:A9,B2:B9) = .3909

Answer 2

The start of solving ANY mathematical problem is to look at your definitions. Math is like a foreign language, you have to know what the words mean before you can put them in a sentence. So start with the basic. What is the mathematical definition of Pierson's correlation coefficient r?

equation 1 see https://www.statisticshowto.com/wp-content/uploads/2009/11/pearsons.gif

where r_xy is the Pierson's r (correlation coefficient for a set of numbers x and y.

x_i is the i'th value of x in a set of n numbers

y_i is the i'th value of y in a set of n numbers

Ok, for our problem the set X = {17, 21, 11, 16, 15, 11, 24, 27}

and Y = {73, 66, 64, 61, 70, 71, 90, 68}

For this equation we only have to calculate xy , x², y² and then sum all the values for x, y, xy, x², y² and use equation 2 to calculate r

For the formula, see video.

Using formula we get r =

or r = 1126 / 2880 = 0.3909