Confidence Intervals for Correlation and Proportion

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

1. Convert $r$ to $z'$
2. Compute a confidence interval in terms of $z'$
3. Convert the confidence interval back to $r$.

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 $ρ$ based on this $r$ of -0.654.

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

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

For this example, $N = 34$ and therefore the standard error is 0.180. The $Z$ for a 95% confidence interval ($Z_{.95}$) 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 $r$ using a calculator. The $r$ associated with a $z'$ of -1.13 is -0.81 and the $r$ associated with a $z'$ of -0.43 is -0.40. Therefore, the population correlation ($ρ$) is likely to be between -0.81 and -0.40. The 95% confidence interval is:

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

Converting back to $r$, 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.

A candidate in a two-person election commissions a poll to determine who is ahead. The pollster randomly chooses 500 registered voters and determines that 260 out of the 500 favor the candidate. In other words, 0.52 of the sample favors the candidate. Although this point estimate of the proportion is informative, it is important to also compute a confidence interval. The confidence interval is computed based on the mean and standard deviation of the sampling distribution of a proportion. The formulas for these two parameters are shown below:

$μp = π$

$\sigma_ p = \sqrt {\dfrac{ \pi (1- \pi) }{N}}$
Since we do not know the population parameter $π$, we use the sample proportion $p$ as an estimate. The estimated standard error of $p$ is therefore

$\S_p = \sqrt {\dfrac{ p (1- p) }{N}}$

We start by taking our statistic ($p$) and creating an interval that ranges ($Z_{.95}$)($s_p$) in both directions, where $Z_{.95}$ is the number of standard deviations extending from the mean of a normal distribution required to contain 0.95 of the area (see the section on the confidence interval for the mean). The value of $Z_{.95}$ is computed with the normal calculator and is equal to 1.96. We then make a slight adjustment to correct for the fact that the distribution is discrete rather than continuous.

Normal Distribution Calculator

$s_p$ is calculated as shown below:

$\S_p = \sqrt {\dfrac{ .52 (1- .52) }{500}} = 0.0223$

R code:

prop.test(260,500,correct=TRUE)
           
	1-sample proportions test with continuity correction
                  
data:  260 out of 500, null probability 0.5
                  
X-squared = 0.722, df = 1, p-value = 0.3955
                  
alternative hypothesis: true p is not equal to 0.5
                  
95 percent confidence interval:
                  
 0.4752277 0.5644604
                  
sample estimates:
                  
   p
                  
0.52