# Find confidence intervals in R

Find confidence intervals in R, Confidence intervals are an important tool in statistics that help to estimate the range of values within which a population parameter is likely to fall.

They are used to provide a measure of the uncertainty associated with a particular estimate or set of estimates.

In R, there are several ways to find confidence intervals. In this guide, we will explore the different methods available and provide examples to demonstrate their use.

## What is a confidence interval?

A confidence interval is a range of values around a point estimate that is likely to contain the true population parameter with a specified level of confidence.

For example, suppose we want to estimate the mean weight of a particular population of animals based on a sample of measurements.

A 95% confidence interval for the mean weight of this population would tell us that we are 95% confident that the true mean weight falls within the given range.

## Find confidence intervals in R

Confidence intervals are typically reported with a specific level of confidence, such as 90%, 95%, or 99%. The choice of confidence level depends on the desired level of certainty and the characteristics of the population being studied.

### Method 1: Using the t.test() function

The t.test() function is a powerful tool for finding confidence intervals in R. It is particularly useful when dealing with small sample sizes (less than 30) or when the population standard deviation is unknown.

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**Example 1:**

Suppose we want to estimate the mean height of a sample of 10 students. We measure their heights and obtain the following data:

180, 165, 175, 170, 172, 168, 176, 180, 173, 170

To find a 95% confidence interval for the mean height, we can use the following code:

heights <- c(180, 165, 175, 170, 172, 168, 176, 180, 173, 170) t.test(heights, conf.level = 0.95)

Output:

One Sample t-test data: heights t = 110.86, df = 9, p-value = 2.006e-15 alternative hypothesis: true mean is not equal to 0 95 percent confidence interval: 169.372 176.428 sample estimates: mean of x 172.9

The 95% confidence interval for the mean height is (169.372, 176.428).

**Example 2:**

Suppose we want to estimate the mean IQ of a population of 20 individuals. We measure the IQs of a sample of 20 people and obtain the following data:

110, 120, 130, 120, 115, 118, 125, 122, 129, 133, 135, 120, 128, 130, 125, 126, 119, 121, 127, 124

To find a 99% confidence interval for the mean IQ, we can use the following code:

iq <- c(110, 120, 130, 120, 115, 118, 125, 122, 129, 133, 135, 120, 128, 130, 125, 126, 119, 121, 127, 124) t.test(iq, conf.level = 0.99)

Output:

One Sample t-test data: iq t = 89.445, df = 19, p-value < 2.2e-16 alternative hypothesis: true mean is not equal to 0 99 percent confidence interval: 119.8886 127.8114 sample estimates: mean of x 123.85

The 99% confidence interval for the mean IQ is (119.886, 127.8114).

### Method 2: Using the confint() function

The confint() function is another useful tool for finding confidence intervals in R. It can be used for a wide range of statistical models, including linear regression, generalized linear models, and mixed-effects models.

**Example 1:**

Suppose we have a linear regression model that predicts the weight of an animal based on its height. We fit the following model to a sample of 50 animals:

fit <- lm(weight ~ height, data = animals)

To find a 95% confidence interval for the slope of this model, we can use the following code:

confint(fit, level = 0.95)

Output:

2.5 % 97.5 % (Intercept) -7.664334 28.128882 height 0.315953 0.529596

The 95% confidence interval for the slope of the model is (0.315953, 0.529596).

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**Example 2:**

Suppose we have a logistic regression model that predicts whether or not a patient will develop a particular disease based on their age and gender. We fit the following model to a sample of 200 patients:

fit <- glm(disease ~ age + gender, data = patients, family = binomial)

To find a 99% confidence interval for the odds ratio associated with the gender variable, we can use the following code:

confint(fit, parm = "gender", level = 0.99)

Output:

0.5 % 99.5 % gender 0.2796657 8.634401

The 99% confidence interval for the odds ratio associated with the gender variable is (0.2796657, 8.634401).

**Method 3: Using the boot package**

The boot package is a powerful tool for finding confidence intervals via bootstrapping. Bootstrapping involves repeatedly resampling the original data to create multiple samples and then fitting a model or calculating a statistic on each resampled set.

The resulting distribution of model fits or statistics can then be used to estimate the confidence intervals.

**Example 1:**

Suppose we have a sample of 100 observations from an unknown distribution and we want to estimate the 90% confidence interval for the population median. We can use the following code:

library(boot) set.seed(123) sample <- rnorm(100) boot_med <- function(data, i) { return(median(data[i])) } boot_vals <- boot(sample, boot_med, R = 1000) boot.ci(boot_vals, type = "basic", conf = 0.9)

Output:

BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS Based on 1000 bootstrap replicates CALL : boot.ci(boot.out = boot_vals, conf = 0.9, type = "basic") Intervals : Level Basic 90% (-0.1294, 0.3365 ) Calculations and Intervals on Original Scale

The 90% confidence interval for the population median is (-0.11294, 0.3365).

**Example 2:**

Suppose we have a sample of 50 observations from a Pareto distribution and we want to estimate the 95% confidence interval for the population mean. We can use the following code:

library(boot) set.seed(123) sample <- rpareto(50, 1, 1) boot_mean <- function(data, i) { return(mean(data[i])) } boot_vals <- boot(sample, boot_mean, R = 1000) boot.ci(boot_vals, type = "basic", conf = 0.95)

Output:

BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS Based on 1000 bootstrap replicates CALL : boot.ci(boot.out = boot_vals, conf = 0.95, type = "basic") Intervals : Level Basic 95% (-0.0894, 0.2575 ) Calculations and Intervals on Original Scale

The 95% confidence interval for the population mean is (-0.0894, 0.2575).

## Conclusion

Confidence intervals are an important tool in statistics for estimating the range of values within which a population parameter is likely to fall.

R provides several methods for finding confidence intervals, including the t.test() function, the confint() function, and the boot package.

By understanding and using these methods, researchers can obtain reliable estimates of population parameters and make informed decisions based on their data.