Question

For this problem, you are provided with both a dataset and a regression equation that relates the predictor and dependent variables. Given the regression equation, calculate by hand the correlation between the two. You must show all your work in order to receive full credit. Use R to check your computations, and attach the R output.

A researcher believes that if people put effort into a charitable cause, they will be more willing to contribute money to that cause. A group of students was asked to work at a Habitat for Humanity building site for part of one day. At the end of their workday, the students were asked to pledge a sum of money to help the family who would move into that house. The number of hours worked and the sum of money pledged are recorded below. (critical r = .878)

Student	Hours Worked	$ Pledged
1	7	18
2	4	3
3	6	8
4	3	1
5	5	7

$ Pledged = -12.1 + 3.9(Hours)

Answer 1

The correlation coefficient is obtained using the formula

$r=\frac{n\left ( \sum XY \right )-\sum X \sum Y}{\sqrt{\left [ n\left ( \sum X^2 \right ) - \left ( \sum X \right )^2 \right ]\left [ n\left ( \sum Y^2 \right ) - \left ( \sum Y \right )^2 \right ]}}$

From the data values,

Student	Hours Worked, X	$ Pledged, Y	X^2	Y^2	XY
1	7	18	49	324	126
2	4	3	16	9	12
3	6	8	36	64	48
4	3	1	9	1	3
5	5	7	25	49	35
SUM	25	37	135	447	224

$r=\frac{5\left ( 224 \right )-25\times 37}{\sqrt{\left [ 5\left ( 135 \right ) - \left ( 25 \right )^2 \right ]\left [ 5\left ( 447 \right ) - \left ( 37 \right )^2 \right ]}}=0.9371$

The R code is given below,

x <- c(7, 4, 6, 3, 5)

y <- c(18, 3, 8, 1, 7)

cor(x, y, method = c("pearson"))

[1] 0.9371102