ASP Statistical Software A STATISTICAL PACKAGE (ASP) For Business, Economics, And The Social Sciences

# Hypothesis Tests

## Single Sample Hypothesis Tests

There are five single sample hypothesis tests available in ASP. You are given the option of calculating the sample statistics for these tests from variables in a data matrix or of entering the sample statistics directly:

• t Test Of Population Mean. The output generated by this option includes the difference between the sample mean and the hypothesized population mean along with the value of the t statistic for the null hypothesis that this difference is equal to zero. The degrees of freedom for this t statistic are listed along with the probability value and the probability value divided by two. The standard error of the estimated difference between the sample mean and the hypothesized population mean is also given.

• Test Of Population Proportion. The output generated by this option includes the difference between the sample proportion and the hypothesized population proportion along with the value of the Z statistic for the null hypothesis that this difference is equal to zero. The Probability value and the probability value divided by two for this statistic are also listed. Finally, the standard error of the estimated difference between the sample proportion and the hypothesized population proportion is given.

• Sign Test For Median (Single Sample). The output generated by this option includes the number of successes or failures, the probability value of this number, and this probability value divided by two.

• Test Of Population Variance. The output generated by this option includes the value of the chi square statistic for the null hypothesis that the value of the population variance is equal to the hypothesized variance along with its degrees of freedom and probability value.

• Runs Test. The output generated by this option includes the Z statistic for the null hypothesis that the set of runs is random, the probability value and the probability value over two for this Z statistic, and the standard error of the number of positive runs.

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## Two Sample Hypothesis Tests

There are nine two sample hypothesis test options on the HYPOTHESIS TESTS menu. You are given the option of calculating the sample statistics for these tests from variables in a data matrix or of entering the sample statistics directly:

• Difference Of Means Tests. The output for this test depends on whether the test samples are independent or contain matched pairs:

• For independent samples the output generated includes the difference between the two sample means along with the value of the Z statistic for the null hypothesis that the population difference is equal to zero. The probability value and the probability value divided by two for this statistic are listed, and an estimate of the standard error of the estimated difference between the two population means is given.

• For matched pairs the output generated includes the value of the t statistic for the null hypothesis that the mean of the differences is equal to zero. The degrees of freedom for this t statistic are also listed along with its probability value and its probability value divided by two. The standard error of the estimated mean of the matched pair differences is given.

• Difference Of Proportions Tests. The output generated by this option includes the difference between the two sample proportions along with the value of the Z statistic for the null hypothesis that this difference is equal to zero. The probability value and the probability value divided by two for this statistic are also listed along with the standard error of the estimated difference between the two population proportions.

• Wilcoxon Rank-Sum Test. The output generated by this option includes the Z statistic for the null hypothesis that the two populations are identical, the probability value and the probability value over two for this statistic, and the standard error of the sum of the ranks.

• Mann-Whitney Test. The output generated by this option includes the Mann-Whitney U statistic for the null hypothesis that the two populations are identical, the probability value and the probability value over two for this statistic, and the standard error of the sum of the ranks.

• Sign Test (Two Sample). The output generated by this option includes the probability value of the number of positive values, given the null hypothesis that the two samples are identical, along with this value divided by two and the standard error of the estimated number of positive values.

• Wilcoxon Signed-Rank Test. The output generated by this option includes the Z statistic for the null hypothesis that the two populations are identical, the probability value and the probability value over two for this statistic, and the standard error of the minimum sum of the negative and positive signed ranks statistic.

• Rank Order Correlation Test. The output generated by this option depends on the hypothesized population correlation coefficient:

• When the hypothesized population correlation coefficient is not equal to 0, the output generated includes the Fisher r to z transform, the Z statistic for the null hypothesis that the correlation between the two variables is equal to the hypothesized population value, the probability value and the probability value over two for this statistic, and the standard error of the estimated correlation coefficient.

• When the hypothesized population correlation coefficient is equal to 0, the output generated includes the t statistic for the null hypothesis that the correlation between the two variables is equal to 0, the probability value and the probability value over two for this statistic, and the standard error of the estimated correlation coefficient.

• Product Moment Correlation Test. The output generated by this option depends on the hypothesized population correlation coefficient:

• When the hypothesized population correlation coefficient is not equal to 0, the output generated includes the Fisher r to z transform, the Z statistic for the null hypothesis that the correlation between the two variables is equal to the hypothesized population value, the probability value and the probability value over two for this statistic, and the standard error of the estimated correlation coefficient.

• When the hypothesized population correlation coefficient is equal to 0, the output generated includes the t statistic for the null hypothesis that the correlation between the two variables is equal to 0, the probability value and the probability value over two for this statistic, and the standard error of the estimated correlation coefficient.

• Equality Of Two Variances Test. The output generated by this option includes the F statistic for the null hypothesis that the two variances are identical along with its numerator and denominator degrees of freedom. The probability value and the probability value divided by two for this statistic are also reported.

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## Multiple Sample Nonparametric Tests

There are four multiple sample nonparametric test options available in ASP:

• Kruskal-Wallis Test. The output generated by this option includes the chi square statistic, along with its degrees of freedom and probability value, for the null hypothesis that the samples come from the same population.

• Friedman Test. The output for this option consists of the chi square statistic, along with its degrees of freedom and probability value, for the null hypothesis that there are no differences in effects of the N treatments on the experimental groups.

• Cochran’s Q Test. The output generated by this option includes the Cochran’s Q statistic, along with its degrees of freedom and probability value, for the null hypothesis that there are no differences in the effects of the N treatments on the experimental groups.

• Kendall’s Coefficient Of Concordance. The output generated by this option includes Kendall’s coefficient of concordance statistic and a chi square statistic, along with its degrees of freedom and probability value, for the null hypothesis that the rankings are random.

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## Goodness Of Fit Test

The output generated by this option includes the chi square statistic for the null hypothesis that the observed values are generated by the distribution from which the expected values are generated along with its degrees of freedom and probability value.

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## Kolmogorov-Smirnov Test

The output from this option includes the Kolmogorov-Smirnov D statistic. The chi square statistic implied by the Kolmogorov-Smirnov D for the null hypothesis that the observed and expected values represent two independent samples that are generated from the same distribution is also given, along with the degrees of freedom for this statistic and its probability value.

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