t-distribution

When $σ$ (population standard deviation is unknown we need to use a new distribution known as the $t$ -distribution

Characteristics of the $t$ -Distribution

The $t$ distribution shares some characteristics of the standard normal distribution and differs from it in others. The t distribution is similar to the standard normal distribution in these ways:

Bell-shaped
symmetric about the mean
mean=median=mode=0 located at center
curve approaches but never touches the $x$ -axis

The $t$ distribution differs from the standard normal distribution in the following ways:

The variance is greater than 1
differs given different degrees of freedom related to sample size
as sample size increases the associated curve gets closer to standard normal curve

The Degrees of Freedom are the number of values that are free to vary after a sample statistic has been computed.

Example: if the mean of 5 values is 10, then 4 of the 5 values are free to vary. But once 4 values are selected, the fifth value must be a specific number to get a sum of 50, since 50 ÷ 5 = 10. Hence, the degrees of freedom are 5 − 1 = 4, and this value tells the researcher which $t$ curve to use.

For the mean the degrees of freedom for a confidence interval is $n - 1$ where $n$ is the sample size.