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8.22 Sampling Theory of Correlation

By using the sampling correlation coefficient, you often have to estimate the population correlation coefficient. To do so one must know the sampling distribution of the coefficient of correlation. If the population coefficient of correlation = 0, this distribution is symmetric and the statistic having student’s ' t ' - distribution can be used. Otherwise the distribution is skewed. For such a case a transformation, given by Fisher, which produces a statistic which is nearly normally distributed is used .

1) In case of large sample ( > 30 ) the standard error of coefficient of correlation ' r '

is calculated by

2) In case of small samples the S. E. of the coefficient of correlation is calculated in the way as in the large samples, with the difference of 1 - r2 , is used. Instead of is used because in the calculation of coefficient of correlation, 2 degree of freedom are lost and therefore n (df) = n - 2. Thus the S. E. of the coefficient of correlation in small samples is given by

The value of ' t ' is calculated by finding out the ratio between the coefficient of correlation and its S. E.

Thus

Example Use ' t ' test to find whether a correlation coefficient of 0.5 is significant if n = 51.

Solution: r = 0.5 and n = 51 then t = will be

From the table t0.05, r = 49 = 2.01

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Thus the calculated value is much higher than table value of 't', then the correlation is significant.

Example A random sample of 15 from a normal population given the coefficient of correlation of - 0.5. is this significant of the existence of correlation of the population ?

Solution: r = -0.5 and n = 15

The df = n - 2 = 15 - 2 = 13

From the table t0.05, r = 13 = 2.16.

Thus t = - 2.07 < 2.16. hence correlation of is not significant to warrant an existence of correlation in the population.

Example What is the least size of a sample that is appropriated to conclude that a correlation coefficient of 0.32 is significantly greater than 0 and 0.05 level ?

Solution : Let the sample size for t0.05 is ' n ', which is least.

Using One-sided test of ' t ' -distribution at 0.05 level,

Example To study the correlation between stature of father and the stature of son, a sample of size 1600, is taken from the population of fathers and sons. The sample study gives the correlation between the two to be 0.80. Within what limits does it hold true for the population ?

Solution:

The correlation in the population probability lies between = r ± 3 S. E. (when the sample is random and simple )

R = S ± S. E.

= 0.8 ± 3 ´ ( 0.009 )

= ( 0.8 - 0.027, 0.8 + 0.027 )

= ( 0.773, 0.827 )

Index

8.1 Population
8.2 Sample
8.3 Parameters and Statistic
8.4 Sampling Distribution
8.5 Sampling Error
8.6 Central Limit Theorem
8.7 Critical Region
8.8 Testing of Hypothesis
8.9 Errors in Tesitng of Hypothesis
8.10 Power o a Hypothesis Test
8.11 Sampling of Variables
8.12 Sampling of Attributes
8.13 Estimation
8.14 Testing the Difference Between Means
8.15 Test for Difference Between Proportions
8.16 Two Tailed and one Tailed Tests
8.17 Test of Significance for Small Samples
8.18 Students t-distribution
8.19 Distribution of 't' for Comparison of Two Samples Means Independent Samples
8.20 Testing Difference Between Mens of Two Samples Dependent Samples or Matched Paired Observations
8.21 Chi-Square
8.22 Sampling Theory of Correlation
8.23 Sampling Theory of Regression

Chapter 1