Calculation of Sample Size

To calculate the sample size, we need to use the formula:

n = (Z^2 * p * q) / E^2

where,
Z = Z-value for the desired level of confidence (in this case, we assume 95% confidence level which corresponds to a Z-value of 1.96)
p = Proportion of the population with the characteristic of interest (in this case, the correlation coefficient (R) is used as an estimate of the proportion of the population)
q = 1 - p
E = Probable error of the estimate (given as 0.032)

Calculation of p

As given in the question, R = 0.9. We know that the correlation coefficient (R) is a measure of the strength and direction of the linear relationship between two variables. Hence, we can assume that R is an estimate of the proportion of the population with the characteristic of interest (i.e., the proportion of pairs of observations that are linearly related). Therefore, we can use R as an estimate of p.

p = R = 0.9

q = 1 - p = 1 - 0.9 = 0.1

Calculation of Sample Size (n)

Substituting the values of Z, p, q, and E in the formula for sample size, we get:

n = (1.96^2 * 0.9 * 0.1) / 0.032^2

n = 482.1

Therefore, the sample size required is 482.1 or approximately 483.

Explanation

The probable error is a measure of the precision of the estimate of the population parameter. It represents the amount of error that we can expect in our estimate due to the sampling variability. A smaller probable error indicates a more precise estimate.

In this question, we are given the probable error (0.032) and the correlation coefficient (R=0.9). Using these values, we can calculate the sample size required to estimate the population correlation coefficient with a desired level of confidence (95% in this case).

The sample size is an important consideration in statistical analysis because it affects the precision of the estimate. A larger sample size leads to a more precise estimate, while a smaller sample size leads to a less precise estimate. Therefore, it is important to determine the appropriate sample size based on the characteristics of the population and the desired level of precision.