- Check the conditions for a [latex]t[/latex]-distribution, then use a [latex]t[/latex]-distribution to calculate probabilities when appropriate.
Standardized Statistic
When we calculate a [latex]z[/latex]-score for a statistic using simulation to estimate the mean and standard deviation of the sample mean, we call this a standardized statistic. Mathematically, though, we know the exact formulas for these values.
standardized statistic ([latex]z[/latex])
[latex]z=\dfrac{\bar{x}-[\text{mean of } \bar{x}'s]}{\text{std. deviation of } \bar{x}'s}[/latex] [latex]= \dfrac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}[/latex]
where [latex]\bar{x}[/latex] is the sample mean, [latex]\mu[/latex] is the population mean, [latex]\sigma[/latex] is the population standard deviation, and [latex]n[/latex] is the sample size. The statistic is “standardized” since it is centered to have a mean of [latex]0[/latex] and scaled to have a standard deviation of [latex]1[/latex].
If the population distribution is normal or the sample size is sufficiently large, this standardized statistic will follow a standard normal distribution: a normal distribution with a mean of [latex]0[/latex] and a standard deviation of [latex]1[/latex].