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## Noncentral t Distribution

### Definition

The most general representation of the noncentral t distribution is quite complicated. Johnson and Kotz [60] give a formula for the probability that a noncentral t variate falls in the range [–uu].

I(x|ν,δ) is the incomplete beta function with parameters ν and δ. δ is the noncentrality parameter, and ν is the number of degrees of freedom.

### Background

The noncentral t distribution is a generalization of Student's t distribution.

Student's t distribution with n – 1 degrees of freedom models the t-statistic

where is the sample mean and s is the sample standard deviation of a random sample of size n from a normal population with mean μ. If the population mean is actually μ0, then the t-statistic has a noncentral t distribution with noncentrality parameter

The noncentrality parameter is the normalized difference between μ0 and μ.

The noncentral t distribution gives the probability that a t test will correctly reject a false null hypothesis of mean μ when the population mean is actually μ0; that is, it gives the power of the t test. The power increases as the difference μ0μ increases, and also as the sample size n increases.

### Example

The following commands generate a plot of the noncentral t pdf.

```x = (-5:0.1:5)';
p1 = nctcdf(x,10,1);
p = tcdf(x,10);
plot(x,p,'-',x,p1,'-')
```