t-distribution

When \(\sigma\) (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

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.