Chapter 14 Formulas and Vocabulary

• Efficiency (p. 14-4)
• Nonparametric hypothesis tests (p. 14-3)
• Parametric hypothesis tests (p. 14-3)

• Sign test (p. 14-5)
• $S_{\text{data}}$ (p. 14-7
• $Z_{\text{data}}$ (p. 14-8.
  

  $Z_{\text{data}}=\frac{(S_{\text{data}}+0.5)-\frac{n}{2}}{\frac{\sqrt{n}}{2}}$

• Boxplot criterion for assessing symmetry (p. 14-19
• Symmetry (p. 14-18
• $T_{\text{data}}$ (p. 14-22
• Wilcoxon signed rank test (p. 14-20

• Independent samples (p. 14-29
• Wilcoxon rank sum test (p. 14-29

• $Z_{\text{data}}$ (p. 14-31.

  $Z_{\text{data}}=\frac{R_1-{\mu }_R}{{\sigma }_R}$

  where

  $\begin{array}{l}{\mu }_R=\frac{n_1(n_1+n_2+1)}{2}\\ {\sigma }_R=\sqrt{\frac{n_1{n}_2(n_1+n_2+1)}{12}}\end{array}$

  and where $n_{\text{1}}$ and $n_{\text{2}}$ represent the sample sizes for samples 1 and 2, respectively, and $R_{\text{1}}=\text{the sum of the ranks for the first sample}$.

• Kruskal-Wallis test (p. 14-36

• ${\chi }_{\text{data}}^2$ (p. 14-39. The test statistic for the Kruskal-Wallis test is

  ${\chi }_{\text{data}}^2=\frac{12}{N(N+1)}\left(\frac{R_1^2}{n_1}+\frac{R_2^2}{n_2}+\cdots +\frac{R_k^2}{n_k}\right)-3(N+1)$

  where

  • $R_{\text{1}}=\text{the sum of the ranks for the first sample}$
  • $R_{\text{2}}=\text{the sum of the ranks for the second sample}$, and so on, until
  • $R_k=\text{the sum of the ranks for the}k\text{th (last) sample}$

  and where $n_1,n_2,\dots ,n_k$ represent the sample sizes for samples $1,2,\dots ,k$, respectively, and $N=\text{the total number of data values in all the samples combined}$; that is, $N=n_1+n_2+\cdots +n_k$.

• Rank correlation test (p. 14-44

• $r_{\text{data}}$ (p. 14-47. The test statistic for the rank correlation test is

  $r_{\text{data}}=1-\frac{6\Sigma d^2}{n(n^2-1)}$

  where $d=\text{the difference in ranks between the two samples}$.

• $G_{\text{data}}$ (p. 14-57
• Run (p. 14-55
• Runs test for randomness (p. 14-55
• Sequence (p. 14-55

• $Z_{data}$ (p. 14-57. The test statistic for the large-sample case for the runs test for randomness is

  $Z_{\text{data}}=\frac{G-{\mu }_G}{{\sigma }_G}$

  where

  $\begin{array}{ccc}{\mu }_G=\frac{2n_1{n}_2}{n_1+n_2}+1& \text{and}& {\sigma }_G=\sqrt{\frac{2\left(n_1{n}_2\right)(2n_1{n}_2-n_1-n_2)}{{(n_1+n_2)}^2(n_1+n_2-1)}}\end{array}$