# Confidence Intervals for Correlation and Proportion

Site: | Saylor Academy |

Course: | MA121: Introduction to Statistics |

Book: | Confidence Intervals for Correlation and Proportion |

Printed by: | Guest user |

Date: | Monday, November 11, 2024, 6:20 PM |

## Description

First, this section shows how to compute a confidence interval for Pearson's correlation. The solution uses Fisher's z transformation. Then, it explains the procedure to compute confidence intervals for population proportions where the sampling distribution needs a normal approximation.

## 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.

### Video

### Questions

**Question 1 out of 3.**

(0.3, 0.5)

(-0.85, -0.47)

**Question 2 out of 3.**

(.093, .877)

(.058, .687)

**Question 3 out of 3.**

### Answers

- All of them are possible except for (0.72, 1.2). The population correlation cannot be above 1.
- The corresponding z' for r = .45 is .485. The standard error = 1/sqrt(28-3) = .20. The Z for a 95% confidence interval is 1.96. Thus, the upper limit of the confidence interval is .485 + (1.96)(.20). You get .877. The lower limit of the confidence interval is .485 - (1.96)(.20). You get .093. Convert back to r and you get (.093, .705).
- The corresponding z' for r = -.8 is -1.099. The standard error = 1/sqrt(40-3) = .164. The Z for a 99% confidence interval is 2.58. Thus, the upper limit of the confidence interval is -1.099 + (2.58)(.164). You get -.676. Convert back to r and you get -.589.

## Proportion

#### Learning Objectives

- Estimate the population proportion from sample proportions
- Apply the correction for continuity
- Compute a confidence interval

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:

Since we do not know the population parameter , we use the sample proportion as an estimate. The estimated standard error of is therefore

We start by taking our statistic () and creating an interval that ranges ()() in both directions, where 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 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

Lower limit:

Upper limit:

Since the interval extends 0.045 in both directions, the margin of error is 0.045. In terms of percent, between 47.5% and 56.5% of the voters favor the candidate and the margin of error is 4.5%. Keep in mind that the margin of error of 4.5% is the margin of error for the percent favoring the candidate and not the margin of error for the difference between the percent favoring the candidate and the percent favoring the opponent. The margin of error for the difference is 6.36%, the square root of 2 times the margin of error for the individual percent. Keep this in mind when you hear reports in the media; the media often get this wrong.

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

### Video

### Questions

**Question 1 out of 3.**

The estimate of the population proportion is slightly biased, and we need to correct for it.

**Question 2 out of 3.**

**Question 3 out of 3.**

### Answers

- We make these corrections because we approximate a discrete distribution with a continuous one.
- Because the confidence interval ranges from 51% to 59%, the newspaper must have found that 55% of their sample prefer Candidate A.
- Because the confidence interval extends 4% in both directions, the margin of error is 4%. The standard error is sqrt[(.4)(.6)/100] = .0490. The correction = .5/100 = .005. Thus, the upper limit of the 95% confidence interval is: .4 + (1.96)(.049) + .005 = .501. Clearly, there is a large margin of error.