Large Sample Tests - Part 4

 

VII. Testing of single correlation coefficient:

Let 'n' be a sample size and 'r' be the sample correlation coefficient of a bivariate data and 'rho' be the population correlation coefficient. we frame the null hypothesis.

OR

H_0: Sample is drawn from the population ~ H_1: Sample is not drawn from the population.

To test H_0 we use Z statistic.

Case(i) When 

Case(ii) When

Where Z and Zi are the fisher's Z - transformation values defined as.

We compare |Z cal| with the table value Z alpha. If |Z cal| < Z alpha then we accept H_0 and If | Z cal| > Z alpha then we reject H_0.

Problems:

1. A bivariate random sample of 1600 observations has correlation coefficient 0.5. To test whether sample is drawn from population. Whose correlation coefficient is 0.6.

Given that,

We frame the null hypothesis:

OR

H_0: Sample is drawn from the population ~ H_1: Sample is not drawn from the population.

Since, rho < 0.7

To test H_0 we use the Z statistic.

Since |Z cal| = 6.25 > 1.96, we reject H_0 at 5% level of significance. Sample is not drawn from the population.

2. A bivariate random sample of 84 observations has a correlation coefficient 0.70. Test whether sample is drawn from population whose correlation coefficient is 0.80 also find 95% confidence interval of the population correlation coefficient.

Given that,

We frame the null hypothesis:

OR

H_0: Sample is drawn from the population ~ H_1: Sample is not drawn from the population.

To test H_0 we use the  z statistics.

Where,

Hence, we reject H_0 at 5% level of significance.

To find the 95% confidence interval of rho we use the condition |Zcal| < 1.96

We know that,

The 95% confidence interval of population correlation coefficient is 0.5713 < 0.7951.

VIII. Testing the difference between the correlation coefficients.

Let us consider two bi-variate random samples of size n1 and n2.

Let r1 and r2 be the sample correlation coefficients.

We frame the null hypothesis:

OR

H_0: Two samples are drawn from the same population ~ H_1: Two samples are not drawn from the same population. 

To test H_0 we use the Z statistics

We compare |Z cal| with the table value Z alpha, If |Z cal| < Z alpha then we accept H_0. If |Z cal| > Z alpha then we reject H_0.

Problems:

1. Two random samples of sizes 56 and 70 observations have the correlation coefficient 0.72 and 0.80. Test whether two samples are drawn from same population.

Given that,

We frame the null hypothesis:

OR

H_0: Two samples are drawn from the same population ~ H_1: Two samples are not drawn from the same population.

To test H_0 we use the Z Statistic

Since, |Z cal| < 1.96 then we accept H_0 at 5% level of significance. Two samples are drawn from the same population.

2. The sample correlation coefficient is of two bivariate random samples 80 and 100 are 0.45 and 0.38. Test whether two samples are drawn from the same population.

Given that,

We frame the null hypothesis:

OR

H_0: Two samples are drawn from the same population ~ H_1: Two samples are not drawn from the same population.

To test H_0 we use the Z statistic.

Since |Z cal| < 1.96 then we accept the H_0 at 5% level of significance. Two samples are drawn from the same population.

Thank You!