Reading and using a t-distribution table in R Programming Language involves understanding the critical values of the t-distribution, which are commonly used in hypothesis testing and confidence interval estimation. T-distribution is used when dealing with small sample sizes or when the population standard deviation is unknown. Here, we’ll explore how to read and use the t-distribution in R.

Here are several methods to find the critical value (quantile) of the t-distribution so we will discuss all of them.

Using the qt Function

The qt function in R can be used to find the critical value (quantile) of the t-distribution for a given confidence level and degrees of freedom. The function has the following syntax:

qt(p, df, lower.tail = TRUE)

where:

p: The probability for which you want to find the quantile.

df: Degrees of freedom.

lower.tail: If TRUE, probabilities are P(X ≤ x), otherwise, P(X > x).

Finding the Critical Value

Suppose you want to find the critical value of the t-distribution for a 95% confidence level with 10 degrees of freedom.

alpha <- 0.05
df <- 10
# Critical value for 95% confidence level
t_critical <- qt(1 - alpha/2, df)
t_critical

Output:

[1] 2.228139

This gives the critical value for a two-tailed test. For a one-tailed test, you would use qt(1 – alpha, df).

Using the pt Function

The pt function gives the cumulative probability associated with a specific t-value. It has the following syntax:

pt(q, df, lower.tail = TRUE)

where:

q: The quantile (t-value) for which you want the cumulative probability.

df: Degrees of freedom.

lower.tail: If TRUE, probabilities are P(X ≤ x), otherwise, P(X > x).

Finding the Cumulative Probability

Suppose you have a t-value of 2.228 with 10 degrees of freedom and want to find the cumulative probability.

t_value <- 2.228
df <- 10
# Cumulative probability
cumulative_prob <- pt(t_value, df)
cumulative_prob

Output:

[1] 0.9749941

Using the rt Function

The rt function generates random numbers following a t-distribution. This is useful for simulations. The function has the following syntax:

rt(n, df)

where:

n: Number of random observations to generate.

df: Degrees of freedom.

Generating Random Numbers

Generate 1000 random numbers from a t-distribution with 10 degrees of freedom:

df <- 10
n <- 1000
# Generate random numbers
random_numbers <- rt(n, df)
hist(random_numbers, breaks = 30, main = "Histogram of t-distribution", 
     xlab = "t-values")

Output:

t Distribution Table in R

Conclusion

Reading and using the t-distribution in R involves using functions like qt, pt, and rt to find critical values, and cumulative probabilities, and to generate random samples. These tools are essential for conducting t-tests, estimating confidence intervals, and performing various statistical analyses.