Question

You wish to determine if there is a linear correlation between the two variables at a significance level of α=0.01. You have the following bivariate data set.

x	y
75	22.4
50.4	41.3
94.4	27.3
60.1	35
63.9	32.2
43.8	52.1
59	42.2
54.5	35.9
75.8	19.2
49.4	46.4
42.5	59.4
29.7	59.3
65.2	27.9
56.6	31.2
38.6	48.3
50.7	32.9
60	35.1
61	35.6
59.2	32.7
94.4	11.8
42.3	47.5
71.1	19.3
65.9	42.2
17.3	65.3

What is the correlation coefficient for this data set?
r =

To find the p-value for a correlation coefficient, you need to convert to a t-score:

t=√r2(n−2)1−r2

This t-score is from a t-distribution with n–2 degrees of freedom.

What is the p-value for this correlation coefficient?
p-value =

Your final conclusion is that...

• There is insufficient sample evidence to support the claim the there is a correlation between the two variables.
• There is sufficient sample evidence to support the claim that there is a statistically significant correlation between the two variables.

Note: In your calculations, round both r and t to 3 decimal places in ALL calculations.

Answer 1

Using R code as follow we get

x=c(75,50.4,94.4,60.1,63.9,43.8,59,54,75.8,49.4,42.5,29.7,65.2,56.6,38.6,50.7,60,61,59.2,94.4,42.3,71.1,65.9,17.3)
> y=c(22.4,41.3,27.3,35,32.2,52.1,42.2,35.9,19.2,46.4,59.3,59.4,27.9,31.2,48.3,32.9,35.1,35.6,32.7,11.8,47.5,19.3,42.2,65.3)
> cor(x,y)
[1] -0.8912305

Correlation coefficient for data is r= - 0.892

Ho : Correlation coefficent is zero

v/s

H1: correlation coefficient is not equal to zero

> cor.test(x,y,conf.level = 0.99)

   Pearson's product-moment correlation

data: x and y
t = -9.2167, df = 22, p-value = 5.212e-09
alternative hypothesis: true correlation is not equal to 0
99 percent confidence interval:
-0.9633119 -0.6992257
sample estimates:
cor
-0.8912305

Here

t= -0.923

Looking in t table we get -0.92167 value is close to zero at 0.01

and R out put alos show that P value is close to zero = zero

P-value=5.212e-09 0

Decision Rule:- P value < 0.01 we reject Ho

Conclusion : There is sufficient sample evidence to support that claim that there is statistically significant correlation between two variables