Module org.bzdev.math

Package org.bzdev.math.stats

Class StudentsTStat.Mean1

java.lang.Object

org.bzdev.math.stats.Statistic

org.bzdev.math.stats.StudentsTStat

org.bzdev.math.stats.StudentsTStat.Mean1

Enclosing class:

StudentsTStat

public static class StudentsTStat.Mean1
extends StudentsTStat

Student's t-test for determining if the mean value of a data set has a specified value.

For a data set X, the statistic is $t = \frac{\mbox{x̅} - \mu_0}{(s_X/\sqrt{n})}$ where

• x is the mean value of the data set X.
• μ₀ is the value against which the mean of X is to be tested.
• s_X is the sample standard deviation of X.
• n is the size of the data set X.

Nested Class Summary

Nested classes/interfaces inherited from class org.bzdev.math.stats.StudentsTStat

StudentsTStat.Mean1, StudentsTStat.Mean2, StudentsTStat.PairedDiff, StudentsTStat.Slope

Nested classes/interfaces inherited from class org.bzdev.math.stats.Statistic

Statistic.PValueMode

Constructor Summary

Constructors

Constructor	Description
Mean1(double mu)	Constructor.
Mean1(double mu, double[] array)	Constructor providing a data set.
Mean1(double mu, double mean, double variance, long n)	Constructor given a mean value to test against and a description of a data set.

Method Summary

Modifier and Type	Method	Description
void	add(double y)	Add additional data.
double	getMean()	Get the mean value of the data set.
double	getNCParameter(double diff)	Get the noncentrality parameter given a difference in mean values.
double	getSDev()	Get the sample standard deviation of the data set.
double	getValue()	Get the value of this statistic.

Methods inherited from class org.bzdev.math.stats.StudentsTStat

getDegreesOfFreedom, getDistribution, getDistribution, setDegreesOfFreedom

Methods inherited from class org.bzdev.math.stats.Statistic

getBeta, getBeta, getCriticalValue, getNCParameter, getPower, getPower, getPValue, optimalValue

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

Mean1

public Mean1(double mu)

Constructor.

Parameters:

mu - the specified mean value

Mean1

public Mean1(double mu,
 double mean,
 double variance,
 long n)

Constructor given a mean value to test against and a description of a data set.

Parameters:

mu - the mean value to test against

mean - the mean value of the data set

variance - the sample variance of the data set

n - the size of the data set

Mean1

public Mean1(double mu,
 double[] array)

Constructor providing a data set.

Parameters:

mu - the specified mean value

array - the data whose mean is to be tested

Method Details

add

public void add(double y)

Add additional data.

Parameters:

y - an additional data point

getMean

public double getMean()

Get the mean value of the data set.

Returns:

the mean value

getSDev

public double getSDev()

Get the sample standard deviation of the data set.

Returns:

the sample standard deviation

getValue

public double getValue()

Description copied from class: Statistic

Get the value of this statistic.

Specified by:

getValue in class Statistic

Returns:

the value of this statistic

getNCParameter

public double getNCParameter(double diff)

Description copied from class: StudentsTStat

Get the noncentrality parameter given a difference in mean values. A Student's t-test can typically be written as $$ T = \frac{\mbox{x̅} - \mu_0}{ \frac{\sigma}{\sqrt{n}} \sqrt{\frac{S}{\sigma^2\nu}} }$$ where ν is the number of degrees of freedom, σ is the population standard deviation of a random variable for which x is the sample mean of X, μ₀ is the population mean of X, n is the sample size, and S is a sum of squares. Note that $\sqrt{\frac{S}{\sigma^2\nu}}$ will be the sample standard deviation for a suitable choice of ν.

If we set θ = μ₁ - μ₀, we can express T as $$T = \frac{\mbox{x̅} - \mu_0 +\theta - \theta }{ \frac{\sigma}{\sqrt{n}} \frac{S/\sigma^2}{\sqrt{\nu}} }$$ or $$T = \frac{\mbox{x̅} - \mu_1 + \theta }{ \frac{\sigma}{\sqrt{n}} \frac{S/\sigma^2}{\sqrt{\nu}} } \ .$$ If we set $Z = \frac{\mbox{x̅} - \mu_1}{\sigma/\sqrt{n}}$ and $\mu = \frac{\theta}{\sigma/\sqrt{n}}$, then T can be written as $\frac{Z+\mu}{\sqrt{V/\nu}}$. If the actual mean is μ₁, then Z has a normal distribution with a mean of zero and a variance of 1. Meanwhile V has a χ² distribution with ν degrees of freedom, and Z and V are independent. Consequently, the random variable $\frac{Z + \mu}{\sqrt{V/\nu}}$ has a noncentral t distribution characterized by the number of degrees of freedom ν and the noncentrality parameter μ.

For citations for specific cases, please see

• Sample Size & Power Calculations and Sample size and power calculations using the noncentral t-distribution for two-sample case.
• Statistical power of the t-tests for the one-sample case.

The notation in this documentation will differ slightly due to the use of a lower-case x instead of an upper-case X as part of a work-around for a mathjax bug involving the LaTeX 'overline' command.

Specified by:

getNCParameter in class StudentsTStat

Parameters:

diff - the difference of the H1 mean value and the H0 mean value.

Returns:

the noncentrality parameter