# Confidence Intervals for Correlation and Proportion

## Correlation

#### Learning Objectives

The computation of a confidence interval on the population value of Pearson's correlation () is complicated by the fact that the sampling distribution of r is not normally distributed. The solution lies with Fisher's transformation described in the section on the sampling distribution of Pearson's . The steps in computing a confidence interval for are:

Let's take the data from the case study Animal Research as an example. In this study, students were asked to rate the degree to which they thought animal research is wrong and the degree to which they thought it is necessary. As you might have expected, there was a negative relationship between these two variables: the more that students thought animal research is wrong, the less they thought it is necessary. The correlation based on 34 observations is -0.654. The problem is to compute a 95% confidence interval on based on this of -0.654.

The conversion of to can be done using a calculator. This calculator shows that the associated with an of -0.654 is -0.78.

The sampling distribution of is approximately normally distributed and has a standard error of

For this example, and therefore the standard error is 0.180. The for a 95% confidence interval () is 1.96, as can be found using the normal distribution calculator (setting the shaded area to .95 and clicking on the "Between" button). The confidence interval is therefore computed as:

The final step is to convert the endpoints of the interval back to using a calculator. The associated with a of -1.13 is -0.81 and the associated with a of -0.43 is -0.40. Therefore, the population correlation () is likely to be between -0.81 and -0.40. The 95% confidence interval is:

To calculate the 99% confidence interval, you use the for a 99% confidence interval of 2.58 as follows:

Converting back to , the confidence interval is:

Naturally, the 99% confidence interval is wider than the 95% confidence interval.

##### R code:

install.packages("psychometric") library(psychometric) CIr(r=-.654, n = 34, level = .95)

[1] -0.8124778 -0.4055190

CIr(r=-.654, n = 34, level = .99)

[1] -0.8468443 -0.3091669

Source: David M. Lane, https://onlinestatbook.com/2/estimation/correlation_ci.html

This work is in the Public Domain.