A study about the productivity of the workers (random sample of 25 employees) in a factory has been completed. The correlation ratio between the productivity of the workers and their educations is r=0.969. Since it is obtained on the basis of the sample can be considered as
point estimate of the coefficient of the general population (with stochastic error). Find a 95% confidence interval (using Fisher). Please, show your full solution!
To find the 95% confidence interval using Fisher's transformation, we need to follow these steps:
Step 1: Determine the sample size
In this case, the sample size is given as 25 employees.
Step 2: Calculate the Fisher's transformation
The Fisher's transformation is denoted as z = 0.5 * ln((1+r)/(1-r)). Plugging in the given correlation coefficient value (r = 0.969), we can calculate the Fisher's transformation as follows:
z = 0.5 * ln((1 + 0.969)/(1 - 0.969))
= 0.5 * ln(1.969/0.031)
= 0.5 * ln(63.548)
= 0.5 * 4.151
= 2.0755
Step 3: Calculate the standard error
The standard error is given by SE = 1/sqrt(n - 3), where n is the sample size. Plugging in the given sample size (n = 25), we can calculate the standard error as follows:
SE = 1/sqrt(25 - 3)
= 1/sqrt(22)
= 1/4.6904
= 0.2132
Step 4: Calculate the 95% confidence interval limits
The confidence interval limits are given by z ± (1.96 * SE), where z is the Fisher's transformation and SE is the standard error. Plugging in the calculated values, we can calculate the confidence interval limits as follows:
Lower limit = 2.0755 - (1.96 * 0.2132)
= 2.0755 - 0.4181
= 1.6574
Upper limit = 2.0755 + (1.96 * 0.2132)
= 2.0755 + 0.4181
= 2.4936
Step 5: Interpret the results
The 95% confidence interval for the correlation coefficient is (1.6574, 2.4936). This means that we are 95% confident that the true correlation coefficient between productivity and education in the general population falls within this range.