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Testing Variances
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Testing Variances
The one sample chi-square test is used to test the hypothesis
that the population from which a sample is drawn has a specified
variance or standard deviation. The F test compares the variances
of two samples. Other tests, such as Levene's test and Bartlett's
test, are used to compare the variances of multiple samples. They
are covered in their own section on testing equality of
variances.
The One Sample Chi-Square Test
The one sample chi-square test is used to test the hypothesis
that a sample comes from a population with a specified variance or
standard deviation. The test is based on the assumption that the
sample is randomly selected from the population, and that the
population itself follows a normal distribution. If either of these
assumptions is violated, the reliability of the chiSquare
test may be compromised.
The null hypothesis is always that the population underlying the
sample has a variance that is equal to the proposed variance. The
alternative hypothesis depends on whether a one or two-tailed test
is performed.
For the one-tailed test, the alternative hypothesis is that the
population from which the sample was drawn has a variance that is
either less than (lower tailed) or greater than (upper tailed) the
proposed variance. For the two-tailed version, the alternative
hypothesis is that the variance of the population does not equal
the the proposed variance.
This test should not be confused with the chi-square
goodness-of-fit test, which is used to verify that a sample comes
from a specified distribution.
The one sample chi-square test is implemented by the
OneSampleChiSquareTest
class. It has five constructors. The first constructor takes no
arguments. The data and conditions for the test must be specified
by setting properties of the OneSampleChiSquareTest
object.
The remaining four constructors can be divided into two
pairs.
The second and third constructors take 2 or 3 arguments. The
first argument is a NumericalVariable
that specifies the sample. The second argument is the proposed
variance. The third, optional argument is a HypothesisType
value that specifies whether the test is one or two-tailed. The
default value is HypothesisType.TwoTailed.
The last two constructors take 3 or 4 arguments. The first
argument is an integer that specifies the size of the sample. The
second argument specifies the variance of the sample. The third
parameter specifies the variance to test again. The optional fourth
argument is a HypothesisType
value that specifies whether the test is one or two-tailed. The
default value is HypothesisType.TwoTailed.
Example
The test scores of a class on a national test are as
follows:
61, 77, 61, 90, 72, 51, 75, 83, 53, 82, 82, 66, 68, 57, 61, 61,
78, 69, 65.
We want to investigate if the variance of the scores is greater
than 5. The following code performs the test:
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double[] group1Data = new double[]
{62, 77, 61, 94, 75, 82, 86, 83, 64, 84,
68, 82, 72, 71, 85, 66, 61, 79, 81, 73};
NumericalVariable results = new NumericalVariable("Class 1", group1Data);
OneSampleChiSquareTest chiSquareTest = new OneSampleChiSquareTest();
chiSquareTest.Sample = results;
chiSquareTest.Variance = 5;
Console.WriteLine("Test statistic: {0:F4}", chiSquareTest.Statistic);
Console.WriteLine("P-value: {0:F4}", chiSquareTest.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
chiSquareTest.Reject() ? "yes" : "no"); |
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Dim group1Data As Double() = New Double() _
{62, 77, 61, 94, 75, 82, 86, 83, 64, 84,
68, 82, 72, 71, 85, 66, 61, 79, 81, 73}
Dim group1Results As NumericalVariable = _
New NumericalVariable("Class 1", group1Data)
Dim chiSquareTest As OneSampleChiSquareTest = _
New OneSampleChiSquareTest()
chiSquareTest.Sample = results
chiSquareTest.Variance = 5
Console.WriteLine("Test statistic: {0:F4}", chiSquareTest.Statistic)
Console.WriteLine("P-value: {0:F4}", chiSquareTest.PValue)
Console.WriteLine("Reject null hypothesis? {0}", _
IIf(chiSquareTest.Reject(), "yes", "no")) |
The value of the chi square statistic turns out to be -2.4505
giving a p-value of 0.0143. As a result, the hypothesis that on
average, the students in this class score no different than the
national average is rejected at the 0.05 level.
Using pre-calculated values for the mean and sample size, the
above example would look like this:
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double mean = 75.3
int sampleSize = 20;
OneSampleChiSquareTest chiSquareTest = new OneSampleChiSquareTest(mean, sampleSize, 79.3, 7.3);
Console.WriteLine("Test statistic: {0:F4}", chiSquareTest.Statistic);
Console.WriteLine("P-value: {0:F4}", chiSquareTest.PValue);
Console.WriteLine("Reject null hypothesis? {0}",
chiSquareTest.Reject() ? "yes" : "no"); |
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Dim mean As Double = 75.3
Dim sampleSize As Integer = 20
Dim chiSquareTest As OneSampleChiSquareTest = _
New OneSampleChiSquareTest(mean, sampleSize, 79.3, 7.3)
Console.WriteLine("Test statistic: {0:F4}", chiSquareTest.Statistic)
Console.WriteLine("P-value: {0:F4}", chiSquareTest.PValue)
Console.WriteLine("Reject null hypothesis? {0}", _
IIf(chiSquareTest.Reject(), "yes", "no")) |
Once a OneSampleChiSquareTest object has been
created, you can access other properties and methods common to all
hypothesis test classes. For instance, to obtain a 95% confidence
interval around the mean, the code would be:
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Interval meanInterval = chiSquareTest.GetConfidenceInterval();
Console.WriteLine("95% Confidence interval for the mean: {0:F1} - {1:F1}",
meanInterval.LowerBound, meanInterval.UpperBound); |
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Dim meanInterval As Interval = chiSquareTest.GetConfidenceInterval()
Console.WriteLine("95% Confidence interval for the mean: {0:F1} - {1:F1}", _
meanInterval.LowerBound, meanInterval.UpperBound) |
The confidence interval for the mean is 72.1 and 78.5 at the 95%
confidence level.
The F-Test
The F-test is a two sample test that is used to test the
hypothesis that the variances of two populations are equal. The
test is based on the assumption that the sample is randomly
selected from the population, and that the population itself
follows a normal distribution. If either of these assumptions is
violated, the reliability of the chiSquare test may be
compromised.
The null hypothesis is always that the population underlying the
sample has a variance that is equal to the proposed variance. The
alternative hypothesis depends on whether a one or two-tailed test
is performed.
For the one-tailed test, the alternative hypothesis is that the
population from which the sample was drawn has a variance that is
either less than (lower tailed) or greater than (upper tailed) the
proposed variance. For the two-tailed version, the alternative
hypothesis is that the variance of the population does not equal
the the proposed variance.
The F test was named in honor of Sir Ronal Fisher who created
the foundations for much of modern statistical analysis.
The F test is implemented by the FTest class. It has
five constructors.
The first constructor takes no arguments. All test parameters
must be provided by setting the properties of the
FTest object.
The remaining four constructors can be divided into two pairs.
The first pair has 2 or 3 arguments. The first two arguments are
NumericalVariable
objects that represent the samples the test is to be applied to.
The first constructor only has these two arguments. This creates a
two-tailed test for equality of variances. The second constructor
of the pair takes a third parameter: a HypothesisType
value that specifies whether the test is one or two-tailed.
One-tailed F tests are very common.
The second pair of constructors take 4 or 5 arguments. The first
four arguments are, in order, the degrees of freedom and variance
of the numerator, and the degrees of freedom and variance of the
denominator. The fifth parameter, if present, is once again a
HypothesisType
value that specifies whether the test is one or two-tailed.
Example
Once again, we use the same data as before. However, this time
we compare the results of one group of students to the results of a
second group of students, with these test scores:
61, 80, 98, 90, 94, 65, 79, 75, 74, 86, 76, 85, 78, 72, 76, 79,
65, 92, 76, 80
We want to test if the variances of the two populations are
equal. The code below performs this test:
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double[] group2Data = new double[]
{61, 80, 98, 90, 94, 65, 79, 75, 74, 86,
76, 85, 78, 72, 76, 79, 65, 92, 76, 80};
NumericalVariable group2Results = new NumericalVariable("Class 2", group2Data);
FTest fTest = new FTest(group1Results, group2Results);
Console.WriteLine("Test statistic: {0:F4}", fTest.Statistic);
Console.WriteLine("P-value: {0:F4}", fTest.Probability);
Console.WriteLine("Reject null hypothesis? {0}",
fTest.Reject() ? "yes" : "no"); |
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Dim group2Data As Double() = _
{61, 80, 98, 90, 94, 65, 79, 75, 74, 86, _
76, 85, 78, 72, 76, 79, 65, 92, 76, 80}
Dim group2Results As NumericalVariable = _
New NumericalVariable("Class 2", group2Data)
Dim fTest As FTest = New FTest(group1Results, group2Results)
Console.WriteLine("Test statistic: {0:F4}", fTest.Statistic)
Console.WriteLine("P-value: {0:F4}", fTest.PValue)
Console.WriteLine("Reject null hypothesis? {0}",
IIf(fTest.Reject(), "yes", "no")) |
The value of the F-statistic is 0.9573 giving a p-value of
0.5374. As a result, the hypothesis that the variance of the scores
of students from the first group is no different than that of the
second group is not rejected at the 0.05 level.
Up: Hypothesis Tests Next: Testing Goodness-of-Fit Previous: Testing Means Contents
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