13.6: Chapter 13 Formulas

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Ranking Data

Order the data from smallest to largest.
The smallest value gets a rank of 1.
The next smallest gets a rank of 2, etc.
If there are any values that tie, then each of the tied values gets the average of the corresponding ranks.

Sign Test

\(H_{0}:\) Median \(= MD_{0}\)
\(H_{1}:\) Median \(\neq MD_{0}\)
p-value uses binomial distribution with \(p = 0.5\) and \(n\) is the sample size not including ties with the median or differences of 0.

For a two-tailed test, the test statistic, \(x\), is the smaller of the plus or minus signs. If \(x\) is the test statistic, the p-value for a two-tailed test is \(2* \text{P}(X \leq x)\).
For a right-tailed test, the test statistic, \(x\), is the number of plus signs. For a left-tailed test, the test statistic, \(x\), is the number of minus signs. The p-value for a one-tailed test is the \(\text{P}(X \geq x)\) for a right-tailed test, or \(\text{P}(X \leq x)\) for a left-tailed test.

Wilcoxon Signed-Rank Test

\(n\) is the sample size not including a difference of 0.

When \(n < 30\), use test statistic \(w_{s}\), which is the absolute value of the smaller of the sum of ranks. CV uses table in Figure 13-5. If critical value is not in tables then use an online calculator: http://www.socscistatistics.com/tests/signedranks.

When \(n \geq 30\), use z-test statistic: \[z = \frac{\left(w_{s} - \left(\dfrac{n (n+1)}{4}\right) \right)}{\sqrt{\left( \dfrac{n(n+1)(2n+1)}{24} \right)}} \nonumber\]

Mann-Whitney U Test

When \(n_{1} \leq 20\) and \(n_{2} \leq 20\):

\(U_{1} = R_{1} - \frac{n_{1} \left(n_{1}+1\right)}{2}, U_{2} = R_{2} - \frac{n_{2} \left(n_{2}+1\right)}{2}\).

\(U = \text{Min} \left(U_{1}, U_{2}\right)\)

CV uses table in Figures 13-8 or 13-9. If critical value is not in tables then use an online calculator: https://www.socscistatistics.com/tests/mannwhitney/default.aspx.

When \(n_{1} > 20\) and \(n_{2} > 20\) use z-test statistic: \[z = \frac{\left(U - \left(\dfrac{n_{1} \cdot n_{2}}{2}\right) \right)}{\sqrt{\left( \dfrac{n_{1} \cdot n_{2} \left(n_{1} + n_{2} + 1\right)}{12} \right)}} \nonumber\]

“For instance, a race of hyperintelligent pan‐dimensional beings once built themselves a gigantic supercomputer called Deep Thought to calculate once and for all the Answer to the Ultimate Question of Life, the Universe, and Everything. For seven and a half million years, Deep Thought computed and calculated, and in the end announced that the answer was in fact Forty‐two - and so another, even bigger, computer had to be built to find out what the actual question was.”

(Adams, 2002)