Determining confidence interval in R

Output:

[1] 25.33415 29.66585
attr(,"conf.level")
[1] 0.95

Based on the supplied data, the result [1] 25.33415 29.66585 shows the mean’s computed confidence interval. The confidence interval in this instance is [25.33415, 29.66585]. The output of attr(,”conf.level”) shows that 95% confidence was utilized to compute the interval.

Accordingly, based on the sample data provided, we can say with a 95% degree of certainty that the true population means is within the range of 25.33415 to 29.66585.

Calculate for Iris Data Set
Firstly we need to create sample data. R provides inbuilt datasets. In this article, we are going to use the iris dataset for illustration. iris dataset depicts the sepal length, sepal width, petal length, and petal width in centimeters. It provides the data of fifty flowers from each of the three species of iris. The species are:

• Iris setosa
• versicolor
• virginica

Output:

    Sepal.Length Sepal.Width Petal.Length Petal.Width    Species
1            5.1         3.5          1.4         0.2     setosa
2            4.9         3.0          1.4         0.2     setosa
3            4.7         3.2          1.3         0.2     setosa
4            4.6         3.1          1.5         0.2     setosa
5            5.0         3.6          1.4         0.2     setosa
6            5.4         3.9          1.7         0.4     setosa
7            4.6         3.4          1.4         0.3     setosa
8            5.0         3.4          1.5         0.2     setosa
9            4.4         2.9          1.4         0.2     setosa
10           4.9         3.1          1.5         0.1     setosa
11           5.4         3.7          1.5         0.2     setosa
12           4.8         3.4          1.6         0.2     setosa
13           4.8         3.0          1.4         0.1     setosa
14           4.3         3.0          1.1         0.1     setosa
15           5.8         4.0          1.2         0.2     setosa

Method 1: Calculating Intervals using base R

In this method, we will find the confidence interval step-by-step using mathematical formulas and R functions. You can follow the below steps to determine the confidence interval in R.

Step 1: Calculate the mean. The very first step is to determine the mean of the given sample data.

Step 2: Now let’s compute the standard error of the mean. 

In order to compute the standard error of the mean (S_x̄), we need to find the standard deviation (s) and the length of the sample data (n).

Step 3: Determine the t-score that is linked to the confidence level.

In this step, we will compute the t-score related to the confidence level. We are required to have exactly α / 2 probability in the lower and upper tail. R provides the qt() function using which we can calculate the t-score easily. The syntax is given below,

Syntax:

qt(random_variable, degree_of_freedom) 

Parameters:

random_variable: It must be a random variable  

degree_of_freedom: It must be degree of Freedom

Step 4: Compute the margin of error and form the confidence interval.

The margin of error is given by,

t_{α / 2,N – 1} S_x̄

It can be easily calculated as,

 
 The confidence interval is equal to the mean +/- margin of error. It can be calculated as,

Combining all the steps

Example:

Output:

[1] 5.709732 5.976934

The confidence interval in R signifies how much uncertainty is present in statistical data. a fundamental statistical technique, confidence intervals offer a range of likely values for an unknown population parameter based on sample data. They are essential to decision-making, hypothesis testing, and statistical inference.
In other words, it is defined as an interval that depicts a population parameter with a probability of 1 – α. The expression for the confidence interval is given below,

x̄ ± t_{α / 2,N – 1} S_x̄

Here,

x̄ ± t_{α / 2} : It signifies the value required to form an area of α / 2 (each tail of a t-distribution where 

  degree of freedom = n – 1)

S_x̄ = s / √n :  It represents the standard error of the mean