Hypothesis test for correlation

The sample correlation coefficient is useful as a descriptive measure of the strength of linear association in a sample. We can also use the correlation coefficient to test the null hypothesis that there is no linear association in the population between a pair of random variables; that is

We can show that when the null hypothesis is true and the random variable have a joint normal distribution then the random variable

follows a Student’s t distribution with (n- 2) degrees of freedom. The following tests of the null hypothesis

have a significance level of :

1. To test against the alternative

the decision rule is

reject if

2. To test against the alternative

the decision rule is

reject if

3. To test against the two sided alternative

the decision rule is

reject if or

where , and is the number for which

where the random variable follows a Student’s t distribution with (n -2) degrees of freedom.

Example:

A sample data set produced the following information

; ; ; ;

; and

Find the sample correlation, and test against a two sided alternative the null hypothesis that the population correlation is 0. Take .

Solution: Denoting by the population correlation, we want to test

against the two sided alternative

the decision rule is

reject if or

Firstly, let us find the value of sample correlation coefficient

The value of the test statistic is

and

Since we reject . Virtually for any level of we reject hypothesis that there is no association between x and y. These data contain very strong evidence of positive (linear) association between x and y.

Дата добавления: 2015-08-05