When to Use the t-Distribution

The t-distribution is used when the standard deviation of the population is unkown. Instead, you use the standard deviation of the sample to calculate the t-statistic:  \(t = {\bar{x} - \mu \over {s \over \sqrt{n}} }\) 



1. Always centered at 0

2. Only one parameter (degrees of freedom)

3. As the degrees of freedom increases, the width if the density curve decreases (gets closer to standard normal)  

4. Wider confidence interval due to more uncertainty

For the t-distribution, the degrees of freedom = \(n-1\)




1. N ≥ 10n (test for independence).

2. Data collected using simple random sample.

3. Sampling distribution of x-bar is approximately normal (If n is less than 30, you must graph the data to ensure no skewed data/outliers).