org.apache.commons.math.stat.inference

## Interface TTest

• All Known Implementing Classes:
TTestImpl

`public interface TTest`
An interface for Student's t-tests.

Tests can be:

• One-sample or two-sample
• One-sided or two-sided
• Paired or unpaired (for two-sample tests)
• Homoscedastic (equal variance assumption) or heteroscedastic (for two sample tests)
• Fixed significance level (boolean-valued) or returning p-values.

Test statistics are available for all tests. Methods including "Test" in in their names perform tests, all other methods return t-statistics. Among the "Test" methods, `double-`valued methods return p-values; `boolean-`valued methods perform fixed significance level tests. Significance levels are always specified as numbers between 0 and 0.5 (e.g. tests at the 95% level use `alpha=0.05`).

Input to tests can be either `double[]` arrays or `StatisticalSummary` instances.

• ### Method Summary

All Methods
Modifier and Type Method and Description
`double` ```homoscedasticT(double[] sample1, double[] sample2)```
Computes a 2-sample t statistic, under the hypothesis of equal subpopulation variances.
`double` ```homoscedasticT(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)```
Computes a 2-sample t statistic, comparing the means of the datasets described by two `StatisticalSummary` instances, under the assumption of equal subpopulation variances.
`double` ```homoscedasticTTest(double[] sample1, double[] sample2)```
Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays, under the assumption that the two samples are drawn from subpopulations with equal variances.
`boolean` ```homoscedasticTTest(double[] sample1, double[] sample2, double alpha)```
Performs a two-sided t-test evaluating the null hypothesis that `sample1` and `sample2` are drawn from populations with the same mean, with significance level `alpha`, assuming that the subpopulation variances are equal.
`double` ```homoscedasticTTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)```
Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances, under the hypothesis of equal subpopulation variances.
`double` ```pairedT(double[] sample1, double[] sample2)```
Computes a paired, 2-sample t-statistic based on the data in the input arrays.
`double` ```pairedTTest(double[] sample1, double[] sample2)```
Returns the observed significance level, or p-value, associated with a paired, two-sample, two-tailed t-test based on the data in the input arrays.
`boolean` ```pairedTTest(double[] sample1, double[] sample2, double alpha)```
Performs a paired t-test evaluating the null hypothesis that the mean of the paired differences between `sample1` and `sample2` is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0, with significance level `alpha`.
`double` ```t(double[] sample1, double[] sample2)```
Computes a 2-sample t statistic, without the hypothesis of equal subpopulation variances.
`double` ```t(double mu, double[] observed)```
Computes a t statistic given observed values and a comparison constant.
`double` ```t(double mu, StatisticalSummary sampleStats)```
Computes a t statistic to use in comparing the mean of the dataset described by `sampleStats` to `mu`.
`double` ```t(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)```
Computes a 2-sample t statistic , comparing the means of the datasets described by two `StatisticalSummary` instances, without the assumption of equal subpopulation variances.
`double` ```tTest(double[] sample1, double[] sample2)```
Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays.
`boolean` ```tTest(double[] sample1, double[] sample2, double alpha)```
Performs a two-sided t-test evaluating the null hypothesis that `sample1` and `sample2` are drawn from populations with the same mean, with significance level `alpha`.
`double` ```tTest(double mu, double[] sample)```
Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the input array with the constant `mu`.
`boolean` ```tTest(double mu, double[] sample, double alpha)```
Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which `sample` is drawn equals `mu`.
`double` ```tTest(double mu, StatisticalSummary sampleStats)```
Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the dataset described by `sampleStats` with the constant `mu`.
`boolean` ```tTest(double mu, StatisticalSummary sampleStats, double alpha)```
Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which the dataset described by `stats` is drawn equals `mu`.
`double` ```tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2)```
Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances.
`boolean` ```tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2, double alpha)```
Performs a two-sided t-test evaluating the null hypothesis that `sampleStats1` and `sampleStats2` describe datasets drawn from populations with the same mean, with significance level `alpha`.
• ### Method Detail

• #### pairedT

```double pairedT(double[] sample1,
double[] sample2)
throws java.lang.IllegalArgumentException,
MathException```
Computes a paired, 2-sample t-statistic based on the data in the input arrays. The t-statistic returned is equivalent to what would be returned by computing the one-sample t-statistic `t(double, double[])`, with `mu = 0` and the sample array consisting of the (signed) differences between corresponding entries in `sample1` and `sample2.`

Preconditions:

• The input arrays must have the same length and their common length must be at least 2.

Parameters:
`sample1` - array of sample data values
`sample2` - array of sample data values
Returns:
t statistic
Throws:
`java.lang.IllegalArgumentException` - if the precondition is not met
`MathException` - if the statistic can not be computed do to a convergence or other numerical error.
• #### pairedTTest

```double pairedTTest(double[] sample1,
double[] sample2)
throws java.lang.IllegalArgumentException,
MathException```
Returns the observed significance level, or p-value, associated with a paired, two-sample, two-tailed t-test based on the data in the input arrays.

The number returned is the smallest significance level at which one can reject the null hypothesis that the mean of the paired differences is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0. For a one-sided test, divide the returned value by 2.

This test is equivalent to a one-sample t-test computed using `tTest(double, double[])` with `mu = 0` and the sample array consisting of the signed differences between corresponding elements of `sample1` and `sample2.`

Usage Note:
The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here

Preconditions:

• The input array lengths must be the same and their common length must be at least 2.

Parameters:
`sample1` - array of sample data values
`sample2` - array of sample data values
Returns:
p-value for t-test
Throws:
`java.lang.IllegalArgumentException` - if the precondition is not met
`MathException` - if an error occurs computing the p-value
• #### pairedTTest

```boolean pairedTTest(double[] sample1,
double[] sample2,
double alpha)
throws java.lang.IllegalArgumentException,
MathException```
Performs a paired t-test evaluating the null hypothesis that the mean of the paired differences between `sample1` and `sample2` is 0 in favor of the two-sided alternative that the mean paired difference is not equal to 0, with significance level `alpha`.

Returns `true` iff the null hypothesis can be rejected with confidence `1 - alpha`. To perform a 1-sided test, use `alpha * 2`

Usage Note:
The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here

Preconditions:

• The input array lengths must be the same and their common length must be at least 2.
• ` 0 < alpha < 0.5 `

Parameters:
`sample1` - array of sample data values
`sample2` - array of sample data values
`alpha` - significance level of the test
Returns:
true if the null hypothesis can be rejected with confidence 1 - alpha
Throws:
`java.lang.IllegalArgumentException` - if the preconditions are not met
`MathException` - if an error occurs performing the test
• #### t

```double t(double mu,
double[] observed)
throws java.lang.IllegalArgumentException```
Computes a t statistic given observed values and a comparison constant.

This statistic can be used to perform a one sample t-test for the mean.

Preconditions:

• The observed array length must be at least 2.

Parameters:
`mu` - comparison constant
`observed` - array of values
Returns:
t statistic
Throws:
`java.lang.IllegalArgumentException` - if input array length is less than 2
• #### t

```double t(double mu,
StatisticalSummary sampleStats)
throws java.lang.IllegalArgumentException```
Computes a t statistic to use in comparing the mean of the dataset described by `sampleStats` to `mu`.

This statistic can be used to perform a one sample t-test for the mean.

Preconditions:

• `observed.getN() > = 2`.

Parameters:
`mu` - comparison constant
`sampleStats` - DescriptiveStatistics holding sample summary statitstics
Returns:
t statistic
Throws:
`java.lang.IllegalArgumentException` - if the precondition is not met
• #### homoscedasticT

```double homoscedasticT(double[] sample1,
double[] sample2)
throws java.lang.IllegalArgumentException```
Computes a 2-sample t statistic, under the hypothesis of equal subpopulation variances. To compute a t-statistic without the equal variances hypothesis, use `t(double[], double[])`.

This statistic can be used to perform a (homoscedastic) two-sample t-test to compare sample means.

The t-statisitc is

` t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var))`

where `n1` is the size of first sample; ` n2` is the size of second sample; ` m1` is the mean of first sample; ` m2` is the mean of second sample

and `var` is the pooled variance estimate:

`var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1)))`

with `var1```` the variance of the first sample and var2 the variance of the second sample. ```

``` Preconditions: ```

``` The observed array lengths must both be at least 2. ```

``` Parameters: sample1 - array of sample data values sample2 - array of sample data values Returns: t statistic Throws: java.lang.IllegalArgumentException - if the precondition is not met t double t(double[] sample1, double[] sample2) throws java.lang.IllegalArgumentException Computes a 2-sample t statistic, without the hypothesis of equal subpopulation variances. To compute a t-statistic assuming equal variances, use homoscedasticT(double[], double[]). This statistic can be used to perform a two-sample t-test to compare sample means. The t-statisitc is    t = (m1 - m2) / sqrt(var1/n1 + var2/n2) where n1 is the size of the first sample n2 is the size of the second sample; m1 is the mean of the first sample; m2 is the mean of the second sample; var1 is the variance of the first sample; var2 is the variance of the second sample; Preconditions: The observed array lengths must both be at least 2. Parameters: sample1 - array of sample data values sample2 - array of sample data values Returns: t statistic Throws: java.lang.IllegalArgumentException - if the precondition is not met t double t(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws java.lang.IllegalArgumentException Computes a 2-sample t statistic , comparing the means of the datasets described by two StatisticalSummary instances, without the assumption of equal subpopulation variances. Use homoscedasticT(StatisticalSummary, StatisticalSummary) to compute a t-statistic under the equal variances assumption. This statistic can be used to perform a two-sample t-test to compare sample means. The returned t-statisitc is    t = (m1 - m2) / sqrt(var1/n1 + var2/n2) where n1 is the size of the first sample; n2 is the size of the second sample; m1 is the mean of the first sample; m2 is the mean of the second sample var1 is the variance of the first sample; var2 is the variance of the second sample Preconditions: The datasets described by the two Univariates must each contain at least 2 observations. Parameters: sampleStats1 - StatisticalSummary describing data from the first sample sampleStats2 - StatisticalSummary describing data from the second sample Returns: t statistic Throws: java.lang.IllegalArgumentException - if the precondition is not met homoscedasticT double homoscedasticT(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws java.lang.IllegalArgumentException Computes a 2-sample t statistic, comparing the means of the datasets described by two StatisticalSummary instances, under the assumption of equal subpopulation variances. To compute a t-statistic without the equal variances assumption, use t(StatisticalSummary, StatisticalSummary). This statistic can be used to perform a (homoscedastic) two-sample t-test to compare sample means. The t-statisitc returned is    t = (m1 - m2) / (sqrt(1/n1 +1/n2) sqrt(var)) where n1 is the size of first sample; n2 is the size of second sample; m1 is the mean of first sample; m2 is the mean of second sample and var is the pooled variance estimate: var = sqrt(((n1 - 1)var1 + (n2 - 1)var2) / ((n1-1) + (n2-1))) with var1 the variance of the first sample and var2 the variance of the second sample. Preconditions: The datasets described by the two Univariates must each contain at least 2 observations. Parameters: sampleStats1 - StatisticalSummary describing data from the first sample sampleStats2 - StatisticalSummary describing data from the second sample Returns: t statistic Throws: java.lang.IllegalArgumentException - if the precondition is not met tTest double tTest(double mu, double[] sample) throws java.lang.IllegalArgumentException, MathException Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the input array with the constant mu. The number returned is the smallest significance level at which one can reject the null hypothesis that the mean equals mu in favor of the two-sided alternative that the mean is different from mu. For a one-sided test, divide the returned value by 2. Usage Note: The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The observed array length must be at least 2. Parameters: mu - constant value to compare sample mean against sample - array of sample data values Returns: p-value Throws: java.lang.IllegalArgumentException - if the precondition is not met MathException - if an error occurs computing the p-value tTest boolean tTest(double mu, double[] sample, double alpha) throws java.lang.IllegalArgumentException, MathException Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which sample is drawn equals mu. Returns true iff the null hypothesis can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2 Examples: To test the (2-sided) hypothesis sample mean = mu at the 95% level, use tTest(mu, sample, 0.05) To test the (one-sided) hypothesis sample mean < mu at the 99% level, first verify that the measured sample mean is less than mu and then use tTest(mu, sample, 0.02) Usage Note: The validity of the test depends on the assumptions of the one-sample parametric t-test procedure, as discussed here Preconditions: The observed array length must be at least 2. Parameters: mu - constant value to compare sample mean against sample - array of sample data values alpha - significance level of the test Returns: p-value Throws: java.lang.IllegalArgumentException - if the precondition is not met MathException - if an error computing the p-value tTest double tTest(double mu, StatisticalSummary sampleStats) throws java.lang.IllegalArgumentException, MathException Returns the observed significance level, or p-value, associated with a one-sample, two-tailed t-test comparing the mean of the dataset described by sampleStats with the constant mu. The number returned is the smallest significance level at which one can reject the null hypothesis that the mean equals mu in favor of the two-sided alternative that the mean is different from mu. For a one-sided test, divide the returned value by 2. Usage Note: The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The sample must contain at least 2 observations. Parameters: mu - constant value to compare sample mean against sampleStats - StatisticalSummary describing sample data Returns: p-value Throws: java.lang.IllegalArgumentException - if the precondition is not met MathException - if an error occurs computing the p-value tTest boolean tTest(double mu, StatisticalSummary sampleStats, double alpha) throws java.lang.IllegalArgumentException, MathException Performs a two-sided t-test evaluating the null hypothesis that the mean of the population from which the dataset described by stats is drawn equals mu. Returns true iff the null hypothesis can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2. Examples: To test the (2-sided) hypothesis sample mean = mu at the 95% level, use tTest(mu, sampleStats, 0.05) To test the (one-sided) hypothesis sample mean < mu at the 99% level, first verify that the measured sample mean is less than mu and then use tTest(mu, sampleStats, 0.02) Usage Note: The validity of the test depends on the assumptions of the one-sample parametric t-test procedure, as discussed here Preconditions: The sample must include at least 2 observations. Parameters: mu - constant value to compare sample mean against sampleStats - StatisticalSummary describing sample data values alpha - significance level of the test Returns: p-value Throws: java.lang.IllegalArgumentException - if the precondition is not met MathException - if an error occurs computing the p-value tTest double tTest(double[] sample1, double[] sample2) throws java.lang.IllegalArgumentException, MathException Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays. The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the two-sided alternative that they are different. For a one-sided test, divide the returned value by 2. The test does not assume that the underlying popuation variances are equal and it uses approximated degrees of freedom computed from the sample data to compute the p-value. The t-statistic used is as defined in t(double[], double[]) and the Welch-Satterthwaite approximation to the degrees of freedom is used, as described here. To perform the test under the assumption of equal subpopulation variances, use homoscedasticTTest(double[], double[]). Usage Note: The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The observed array lengths must both be at least 2. Parameters: sample1 - array of sample data values sample2 - array of sample data values Returns: p-value for t-test Throws: java.lang.IllegalArgumentException - if the precondition is not met MathException - if an error occurs computing the p-value homoscedasticTTest double homoscedasticTTest(double[] sample1, double[] sample2) throws java.lang.IllegalArgumentException, MathException Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the input arrays, under the assumption that the two samples are drawn from subpopulations with equal variances. To perform the test without the equal variances assumption, use tTest(double[], double[]). The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the two-sided alternative that they are different. For a one-sided test, divide the returned value by 2. A pooled variance estimate is used to compute the t-statistic. See homoscedasticT(double[], double[]). The sum of the sample sizes minus 2 is used as the degrees of freedom. Usage Note: The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The observed array lengths must both be at least 2. Parameters: sample1 - array of sample data values sample2 - array of sample data values Returns: p-value for t-test Throws: java.lang.IllegalArgumentException - if the precondition is not met MathException - if an error occurs computing the p-value tTest boolean tTest(double[] sample1, double[] sample2, double alpha) throws java.lang.IllegalArgumentException, MathException Performs a two-sided t-test evaluating the null hypothesis that sample1 and sample2 are drawn from populations with the same mean, with significance level alpha. This test does not assume that the subpopulation variances are equal. To perform the test assuming equal variances, use homoscedasticTTest(double[], double[], double). Returns true iff the null hypothesis that the means are equal can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2 See t(double[], double[]) for the formula used to compute the t-statistic. Degrees of freedom are approximated using the Welch-Satterthwaite approximation. Examples: To test the (2-sided) hypothesis mean 1 = mean 2 at the 95% level, use tTest(sample1, sample2, 0.05). To test the (one-sided) hypothesis mean 1 < mean 2 , at the 99% level, first verify that the measured mean of sample 1 is less than the mean of sample 2 and then use tTest(sample1, sample2, 0.02) Usage Note: The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The observed array lengths must both be at least 2. 0 < alpha < 0.5 Parameters: sample1 - array of sample data values sample2 - array of sample data values alpha - significance level of the test Returns: true if the null hypothesis can be rejected with confidence 1 - alpha Throws: java.lang.IllegalArgumentException - if the preconditions are not met MathException - if an error occurs performing the test homoscedasticTTest boolean homoscedasticTTest(double[] sample1, double[] sample2, double alpha) throws java.lang.IllegalArgumentException, MathException Performs a two-sided t-test evaluating the null hypothesis that sample1 and sample2 are drawn from populations with the same mean, with significance level alpha, assuming that the subpopulation variances are equal. Use tTest(double[], double[], double) to perform the test without the assumption of equal variances. Returns true iff the null hypothesis that the means are equal can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2. To perform the test without the assumption of equal subpopulation variances, use tTest(double[], double[], double). A pooled variance estimate is used to compute the t-statistic. See t(double[], double[]) for the formula. The sum of the sample sizes minus 2 is used as the degrees of freedom. Examples: To test the (2-sided) hypothesis mean 1 = mean 2 at the 95% level, use tTest(sample1, sample2, 0.05). To test the (one-sided) hypothesis mean 1 < mean 2, at the 99% level, first verify that the measured mean of sample 1 is less than the mean of sample 2 and then use tTest(sample1, sample2, 0.02) Usage Note: The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The observed array lengths must both be at least 2. 0 < alpha < 0.5 Parameters: sample1 - array of sample data values sample2 - array of sample data values alpha - significance level of the test Returns: true if the null hypothesis can be rejected with confidence 1 - alpha Throws: java.lang.IllegalArgumentException - if the preconditions are not met MathException - if an error occurs performing the test tTest double tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws java.lang.IllegalArgumentException, MathException Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances. The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the two-sided alternative that they are different. For a one-sided test, divide the returned value by 2. The test does not assume that the underlying popuation variances are equal and it uses approximated degrees of freedom computed from the sample data to compute the p-value. To perform the test assuming equal variances, use homoscedasticTTest(StatisticalSummary, StatisticalSummary). Usage Note: The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The datasets described by the two Univariates must each contain at least 2 observations. Parameters: sampleStats1 - StatisticalSummary describing data from the first sample sampleStats2 - StatisticalSummary describing data from the second sample Returns: p-value for t-test Throws: java.lang.IllegalArgumentException - if the precondition is not met MathException - if an error occurs computing the p-value homoscedasticTTest double homoscedasticTTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2) throws java.lang.IllegalArgumentException, MathException Returns the observed significance level, or p-value, associated with a two-sample, two-tailed t-test comparing the means of the datasets described by two StatisticalSummary instances, under the hypothesis of equal subpopulation variances. To perform a test without the equal variances assumption, use tTest(StatisticalSummary, StatisticalSummary). The number returned is the smallest significance level at which one can reject the null hypothesis that the two means are equal in favor of the two-sided alternative that they are different. For a one-sided test, divide the returned value by 2. See homoscedasticT(double[], double[]) for the formula used to compute the t-statistic. The sum of the sample sizes minus 2 is used as the degrees of freedom. Usage Note: The validity of the p-value depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The datasets described by the two Univariates must each contain at least 2 observations. Parameters: sampleStats1 - StatisticalSummary describing data from the first sample sampleStats2 - StatisticalSummary describing data from the second sample Returns: p-value for t-test Throws: java.lang.IllegalArgumentException - if the precondition is not met MathException - if an error occurs computing the p-value tTest boolean tTest(StatisticalSummary sampleStats1, StatisticalSummary sampleStats2, double alpha) throws java.lang.IllegalArgumentException, MathException Performs a two-sided t-test evaluating the null hypothesis that sampleStats1 and sampleStats2 describe datasets drawn from populations with the same mean, with significance level alpha. This test does not assume that the subpopulation variances are equal. To perform the test under the equal variances assumption, use homoscedasticTTest(StatisticalSummary, StatisticalSummary). Returns true iff the null hypothesis that the means are equal can be rejected with confidence 1 - alpha. To perform a 1-sided test, use alpha * 2 See t(double[], double[]) for the formula used to compute the t-statistic. Degrees of freedom are approximated using the Welch-Satterthwaite approximation. Examples: To test the (2-sided) hypothesis mean 1 = mean 2 at the 95%, use tTest(sampleStats1, sampleStats2, 0.05) To test the (one-sided) hypothesis mean 1 < mean 2 at the 99% level, first verify that the measured mean of sample 1 is less than the mean of sample 2 and then use tTest(sampleStats1, sampleStats2, 0.02) Usage Note: The validity of the test depends on the assumptions of the parametric t-test procedure, as discussed here Preconditions: The datasets described by the two Univariates must each contain at least 2 observations. 0 < alpha < 0.5 Parameters: sampleStats1 - StatisticalSummary describing sample data values sampleStats2 - StatisticalSummary describing sample data values alpha - significance level of the test Returns: true if the null hypothesis can be rejected with confidence 1 - alpha Throws: java.lang.IllegalArgumentException - if the preconditions are not met MathException - if an error occurs performing the test ```
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