NormTwoSample Class

NormTwoSample Class

Computes statistics for mean and variance inferences using samples from two normal populations.

Inheritance Hierarchy

SystemObject
Imsl.StatNormTwoSample

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class NormTwoSample

<SerializableAttribute>
Public Class NormTwoSample

[SerializableAttribute]
public ref class NormTwoSample

[<SerializableAttribute>]
type NormTwoSample =  class end

The NormTwoSample type exposes the following members.

Constructors

	Name	Description
	NormTwoSample	Constructor to compute statistics for mean and variance inferences using samples from two normal populations.

Top

Methods

	Name	Description
	DowndateX	Removes the observations in x from the first sample.
	DowndateY	Removes the observations in y from the second sample.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)
	Update	Concatenates samples x and y to the samples provided in the constructor.
	UpdateX	Concatenates the values in x to the first sample provided in the constructor.
	UpdateY	Concatenates the values in y to the second sample provided in the constructor.

Top

Properties

	Name	Description
	ChiSquaredTest	The test statistic associated with the chi-squared test for common, or pooled, variances.
	ChiSquaredTestDF	The degrees of freedom associated with the chi-squared test for the common, or pooled, variances.
	ChiSquaredTestNull	The null hypothesis value for the chi-squared test.
	ChiSquaredTestP	The probability of a larger chi-squared associated with the chi-squared test for common, or pooled, variances.
	ConfidenceMean	The confidence level (in percent) for a two-sided interval estimate of the mean of x - the mean of y, in percent.
	ConfidenceVariance	The confidence level (in percent) for two-sided interval estimate of the variances.
	DiffMean	The difference of means for the two samples.
	FTest	The F test value of the F test for equality of variances.
	FTestDFdenominator	The denominator degrees of freedom of the F test for equality of variances.
	FTestDFnumerator	The numerator degrees of freedom of the F test for equality of variances.
	FTestP	The probability of a larger F in absolute value for the F test for equality of variances, assuming equal variances.
	LowerCICommonVariance	The lower confidence limits for the common, or pooled, variance.
	LowerCIDiff	The lower confidence limit for the mean of the first population minus the mean of the second for equal or unequal variances.
	LowerCIRatioVariance	The approximate lower confidence limit for the ratio of the variance of the first population to the second.
	MeanX	The mean of the first sample, x.
	MeanY	The mean of the second sample, y.
	PooledVariance	The Pooled variance for the two samples.
	StdDevX	The standard deviation of the first sample, x.
	StdDevY	The standard deviation of the second sample, y.
	TTest	The test statistic for the Satterthwaite's approximation for equal or unequal variances.
	TTestDF	The degrees of freedom for the Satterthwaite's approximation for t-test for either equal or unequal variances.
	TTestNull	The Null hypothesis value for t-test for the mean.
	TTestP	The approximate probability of a larger t for the Satterthwaite's approximation for equal or unequal variances.
	UnequalVariances	Specifies whether to return statistics based on equal or unequal variances.
	UpperCICommonVariance	The upper confidence limits for the common, or pooled, variance.
	UpperCIDiff	The upper confidence limit for the mean of the first population minus the mean of the second for equal or unequal variances.
	UpperCIRatioVariance	The approximate upper confidence limit for the ratio of the variance of the first population to the second.

Top

Remarks

Class NormTwoSample computes statistics for making inferences about the means and variances of two normal populations, using independent samples in x1 and x2. For inferences concerning parameters of a single normal population, see class NormOneSample.

Let $\mu_1$ and $\sigma _1^2$ be the mean and variance of the first population, and let $\mu_2$ and $\sigma _2^2$ be the corresponding quantities of the second population. The function contains test confidence intervals for difference in means, equality of variances, and the pooled variance.

The means and variances for the two samples are as follows:

$\bar x_1 = \left( {\sum {x_{1i} /n_1 } } \right), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \bar x_2 = \left( {\sum {x_{2i} } } \right)/n_2$

and

$s_1^2 = \sum {\left( {x_{1i} - \bar x_1 } \right)} ^2 /\left( {n_1 - 1} \right), \,\,\,\,\,\,\,\,\,\,\,s_2^2 = \sum {\left( {x_{2i} - {\bar x}_2} \right)} ^2 /\left( {n_2 - 1} \right)$

Inferences about the Means

The test that the difference in means equals a certain value, for example, $\mu_0$ , depends on whether or not the variances of the two populations can be considered equal. If the variances are equal and meanHypothesis equals 0, the test is the two-sample t-test, which is equivalent to an analysis-of-variance test. The pooled variance for the difference-in-means test is as follows:

$s^2 = \frac{{\left( {n_1 - 1} \right)s_1 + \left( {n_2 - 1} \right)s_2 }} {{n_1 + n_2 - 2}}$

The t statistic is as follows:

$t = \frac{{\bar x_1 - \bar x_2 - \mu _0 }} {s\sqrt {{\left( {1/n_1 } \right)} + \left( {1/n_2 } \right)}}$

Also, the confidence interval for the difference in means can be obtained by first assigning the unequal variances flag to false. This can be done by setting the UnequalVariances property. The confidence interval can then be obtained by the LowerCIDiff and UpperCIDiff properties.

If the population variances are not equal, the ordinary t statistic does not have a t distribution and several approximate tests for the equality of means have been proposed. (See, for example, Anderson and Bancroft 1952, and Kendall and Stuart 1979.) One of the earliest tests devised for this situation is the Fisher-Behrens test, based on Fisher's concept of fiducial probability. A procedure used in the TTest, LowerCIDiff and UpperCIDiff properties assuming unequal variances are specified is the Satterthwaite's procedure, as suggested by H.F. Smith and modified by F.E. Satterthwaite (Anderson and Bancroft 1952, p. 83). Set UnequalVariances true to obtain results assuming unequal variances.

The test statistic is

$t' = \left( {\bar x_1 - \bar x_2 - \mu _0 } \right)/s_d$

where

$s_d = \sqrt {\left( {s_1^2 /n_1 } \right) + \left( {s_2^2 /n_2 } \right)}$

Under the null hypothesis of $\mu_1- \mu_2= c$ , this quantity has an approximate t distribution with degrees of freedom df, given by the following equation:

${\rm{df}} = \frac{{s_d^4 }}{{\frac{{\left( {s_1^2 /n_1 } \right)^2 }}{{n_1 - 1}} + \frac{{\left( {s_2^2 /n_2 } \right)^2 }}{{n_2 - 1}}}}$

Inferences about Variances

The F statistic for testing the equality of variances is given by $F = s_{\max }^2 /s_{\min }^2$ , where $s_{\max}^2$ is the larger of and . If the variances are equal, this quantity has an F distribution with and degrees of freedom.

It is generally not recommended that the results of the F test be used to decide whether to use the regular t-test or the modified on a single set of data. The modified (Satterthwaite's procedure) is the more conservative approach to use if there is doubt about the equality of the variances.

Reference

Imsl.Stat Namespace

Other Resources

Example