Understanding the t-distribution

When performing a t-test, you first calculate your t-statistic using the familiar formula:

$$ t = \frac{X - M}{SE} $$

$X$ is the observed value, $M$ is the expected value under the null hypothesis (or population mean), and $SE$ is the standard error. Once you've computed the t-statistic, you then compare it to the so-called critical value, which comes from the relevant t-distribution.

The shape of a t-distribution, and thus the critical value, is determined entirely by its degrees of freedom. To demonstrate this, let's draw some density plots for t-distributions using different degrees of freedom.

This exercise is part of the course

Intro to Statistics with R: Student's T-test

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Exercise instructions

Create a vector x that contains a sequence of length 100 between -4 and 4. See ?seq for help.
Use dt() to generate t-distributions with 4, 6, 8, 10, and 12 degrees of freedom (in that order). The first argument to dt() is the vector of values at which to evalute the t-distribution (x from above) and the second argument (df) is the degrees of freedom.
Plot each of the t-distributions. Once the inital plot() is created, you'll use lines() to plot each additional distribution. The two arguments to lines() are the same as the first two arguments to plot(), except that you'll have to substitute the appropriate y-values. Use the color black for 4 degrees of freedom, red for 6, orange for 8, green for 10, and blue for 12.
Add a legend() to your plot. The legend should be situated at the top right corner of your plot and should have the title "t-distributions". This is done by setting the first argument to "topright" and the title argument to "t-distributions".

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Generate a vector of 100 values between -4 and 4
x <- seq(___, ___, length = ___)

# Simulate the t-distribution
y_1 <- dt(x, df = ___)
y_2 <- dt(x, df = ___)
y_3 <- dt(x, df = ___)
y_4 <- dt(x, df = ___)
y_5 <- dt(x, df = ___)

# Plot the t-distributions
plot(x, y_1, type = "l", lwd = 2, xlab = "t-value", ylab = "Density", 
     main = "Comparison of t-distributions", col = "black")
lines(___, ___, col = "red")
lines(___, ___, col = "orange")
lines(___, ___, col = "green")
lines(___, ___, col = "blue")

# Add a legend
legend(___, c("df = 4", "df = 6", "df = 8", "df = 10", "df = 12"), 
       col = c("black", "red", "orange", "green", "blue"), 
       title = ___, lty = 1)

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