Question

14-E1. Below are some (fictitious) correlations between all pairs of three variables. The means and standard deviations are also give. Imagine that they were obtained from a sample of 35 randomly chosen individuals.

	Income	Education	Age
Income ($1000)	1.0	0.43	0.22
Education (years)	0.43	1.00	-0.58
Age (years)	0.22	-0.58	1.00
Mean	45.2	11.3	47
Stdev	8.5	3	12

1. Which, if any, of these correlations are significantly different from zero?
2. What does "significantly different from zero" mean in simple words?
3. How large a sample of individuals would be required for a correlation of 0.1 to be significant?
4. If the units of Age changed to months then which figures would change and how?

Answer 1

b. Here we need to test the significance of correlation coefficients at    level of significance, i.e. we need to test      against,      at level of significance.

So this is a both tailed size      test.

The test statistic for this test is where    is correlation coefficient and    is the sample size.

We know    .

Assuming , for the given value   ,the critical value for this test is   .

So we conclude that a correlation coefficient is significant if , , i.e. we reject the hypothesis   .

a. (i) Now for the correlation coefficient between Education and Income is   .

Hence this is significantly different from zero.

(ii) For the correlation coefficient between Age and Education is   .

Hence this is not significantly different from zero.

(iii) For the correlation coefficient between Age and Income is   .

Hence this is not significantly different from zero.

c. A infinitely large sample will be required.

d. If the age units are changed to months mean and standard deviation of all the variables will be 12 times of the current values but the vales of correlation coefficients will not be changed.