## APPENDIX C

## Formulae for Hypothesis Testing

**Large sample statistical test***θ*may equal*µ p*, (*µ*_{1}–*µ*_{2}), or (*p*_{1}–*p*_{2})H

_{o}:*θ*=*θ*_{o}H

_{a}:*θ*>*θ*_{o}or H_{a}:*θ*<*θ*_{o}(One-tailed test)

H

_{a}:*θ*≠*θ*_{o}(Two-tailed test)

Reject H

_{o}if*Z*>*Z*_{α}or if*Z*< –*Z*_{α}(One-tailed test)

Reject H

_{o}if*Z*>*Z*_{a/2}or if*Z*< –*Z*_{a/2}(Two-tailed test)

**Small-sample test for a population mean**H

_{o}:*µ*=*µ*_{o}H

_{a}:*µ*>*µ*_{o}or*µ*<*µ*_{o}Reject H

_{o}if*t*>*t*_{α}or if*t*< –*t*_{α}(One-tailed test)

Reject H

_{o}if*t*>*t*_{α/2}or if*t*< –*t*_{α/2}(Two-tailed test)

**Small-sample test for the difference between two means based on independent random samples**H

_{o}:*µ*_{1}–*µ*_{o}=*D*_{o}H

_{a}: One or two-tailed hypothesis determined by the experimenter.Reject H

_{o}if*t*>*t*_{α}or if*t*< –*t*_{α}(One-tailed test)

Reject H

_{o}when*t*>*t*_{α/2}or if*t*< –*t*_{α/2}(Two-tailed test)

**Small-sample statistical test for the difference between two means based on a pair-difference experiment**H

_{o}:*µ*_{1}–*µ*_{2}=*µ*_{d}=*D*_{o}*n*is the number of paired differences andStandard deviation of paired differences.

Alternative hypothesis H

_{a}and*α*are specified by the experimenter.**Test of an hypothesis about a population variance**H

_{o}:*σ*^{2}=*σ*^{2}_{o}H

_{a}:*σ*^{2}≠*σ*^{2}_{o}(Two-tailed test)

H

_{a}:*σ*^{2}>*σ*^{2}_{o}or H_{a}:*σ*^{2}<*σ*^{2}_{o}(One-tailed test)

df = (n − 1)

Reject H

_{o}if*X*^{2}<*X*^{2}_{1–α/2}with (*n*– 1) df(Two-tailed test)

Reject H

_{a}:*σ*^{2}>*σ*^{2}_{o}or*σ*^{2}<*σ*^{2}_{o}(One-tailed test)

**Test of hypothesis about equality of two population variances**H

_{o}:*σ*^{2}_{1}=*σ*^{2}_{2}H

_{a}:*σ*^{2}_{1}≠*σ*^{2}_{2}(Two-tailed test)

Test statistic : F = S

^{2}_{1}/S^{2}_{2}df = (*n*_{1}–1) and (*n*_{2}– 1)Reject H

_{o}if*F*>*F*_{α/2}(Two-tailed test)

**Test of a hypothesis concerning the slope of a line**H

_{o}:*β*_{1}=*β*_{1o}H

_{a}: Specified by the experimenterdf = (

*n*– 2)Reject H

_{o}, if*t*<*t*_{e}.**The Sign Test (***n*< 25)H

_{o}: The two population distributions are identical and*P*(A exceeds B for a given pair) =*p*= .5.H

_{a}: The two population distributions are not identical and*p*≠ .5, orH

_{a}: The population of A/B measurements is shifted to the right of the population of B/A measurements and p > .5*.Test statistic: The number of times that (A – B) was positive

Rejection region: Reject H

_{o}if*X*≤*X*_{L}or*X*≥*X*_{U}Where

*X*and_{L}*X*_{U}are the lower and upper-tailed values of a binomial distribution, for a two-tailed test.Reject H

_{o}if*X*≥*X*_{U}Where

*X*_{U}is the upper-tailed value of a binomial distribution, for one-tailed test.**Sign Test for large samples (***n*≥ 25)H

_{o}:*p*= .5H

_{a}:*p*≠ .5(Two-tailed test)

H

_{a}= 0.5 or H_{a}< 0.5(One-tailed test)

Reject H

_{o}if*z*≥*z*_{α/2}or*z*≤*z*_{α/2}**Mann–Whitney U Test**H

_{o}: The population relative frequency distributions for A and B are identical.H

_{a}: The two population relative frequency distributions are shifted with respect to their relative locations (a two-tailed test), orH

_{a}: The population relative frequency distribution for A is shifted to the right of the relative frequency distribution for population B (one-tailed test).*Test statistic: Use U, the smaller of U

_{A}and U_{B}(two-tailed test).*U*_{A}=*n*_{1}*n*_{2}+*n*_{1}(*n*_{1}+ 1)/2 –*T*, and_{A}*U*_{B}=*n*_{1}*n*_{2}+*n*_{2}(*n*_{2}+ 1)0/2 –*T*_{B}Where

*T*and_{A}*T*are the rank sums for samples A and B, respectively._{B}Use

*U*for a one-tailed test._{A}Reject H

_{o}if*U*≤*U*_{o},Where

*P*(*U*≤*U*_{o}) =*α*/2. (two tailed test).For a one-tailed test and a given value of

*α*, reject H_{o}if*U*_{A}≤*U*o =*α*.**Mann–Whitney U Test for large samples (n**_{1}> 10 and n_{2}> 10)H

_{o}: The population relative frequency distribution for A and B are identical.H

_{a}: The two population relative frequency distributions are not identical two-tailed testH

_{a}: The population relative frequency distribution for A/B is shifted to the right (or left) of the relative frequency distribution for population B/A. (One-tailed test)Let U = U

_{A}.Reject H

_{o}if z ≥ z_{α/2}or z ≤ z_{α/2}(Two-tailed test)

Reject H

_{o}when z < -z_{β}(One-tailed test)

**Kruskal–Wallis test for comparing t, more than 2 population distributions**H

_{o}: Distributions of all populations are identical.H

_{a}: At least two of the*t*relative frequency distributions differ.Reject H

_{o}if H > X^{2}_{α}.**Calculation of the test statistic and the Wilcoxon Signed-Rank Test**- Calculate the differences (
*X*–_{A}*X*) for each of the_{B}*n*pairs. Differences equal to zero are eliminated and the number of pairs*n*, is reduced accordingly. - Rank the absolute values of the differences, assigning 1 to the smallest, 2 to the second smallest, and so on. Tied observations are assigned the average of the ranks that would have been assigned with no ties.
- Calculate the rank sum for the negative differences and label this value T
^{–}. Similarly, calculate T^{+}, the rank sum for the positive differences. - For a two-tailed test we use the smaller of these two quantities, T, as the test statistic to test the null hypothesis that the two population relative frequency histograms are identical.

- Calculate the differences (
**Wilcoxon Signed-Rank for a paired experiment**H

_{o}: The two population relative frequency distributions are identical.H

_{a}: The two population relative frequency distributions are not identical, in a two-tailed testH

_{a}: The relative frequency distribution for population A is shifted to the right (or left) of the relative frequency distribution for population B, in a one tailed test.Test statistic: T, the smaller of the rank sum for positive differences and the rank sum for negative differences.

Reject H

_{o}if*T*≤*T*_{o}, where*T*_{o}is the critical value in tables.**Large-sample Wilcoxon Signed-Rank test for a paired experiment (n ≥ 25)**H

_{o}: The two population relative frequency distributions for A and B are identical.H

_{a}: The two population relative frequency distributions differ in location, in a two-tailed testH

_{a}: The population relative frequency distribution for A is shifted to the right (or left) of the relative frequency distribution for population B – a one tailed test.T can be either T

^{+}or T^{–}.Reject H

_{o}if*z*≥*z*_{α/2}or*z*≤ –*z*_{α/2}(Two-tailed test)

Place all

*α*in one tail of the*z*distribution. To detect a shift in the distribution of the A observations to the right of the distribution of the B observations, let T=T^{+}and reject H_{o}when*z*>*z*_{α}.To detect a shift in the opposite direction, let T=T

^{–}and reject H_{o}if*z*< –*z*_{α}(One-tailed test).**Friedman Test for randomised block designs**H

_{o}: The probability distributions for the*t*treatments (populations) are identical.H

_{a}: At least two of the*t*treatments have different probability distributions.Reject H

_{o}if*χ*^{2}>*χ*^{2}_{α}.**Runs Test**H

_{o}: The sequence of elements call them S’s and F’s, has been produced in a random manner.H

_{a}: The elements have been produced in a non-random sequence (a two-tailed test)H

_{a}: The process is non-random owing solely to overmixing (an upper one-tailed test) or solely to undermixing (a lower one-tailed test).Test statistic:

*R*is the number of runs.Reject H

_{o}if*R*≤*k*_{1}or*R*≥*k*_{2}Where

*P*(*R*≤*k*_{1}) +*P*(*R*≥*k*_{2}) =*α*and*k*_{1}and*k*_{2}are obtained from the tables (Two-tailed test).Reject H

_{o}if*R*≤*k*_{1}, where*P*(*R*≤*k*_{1}) =*α*and*k*_{1}is obtained from the tables (Two-tailed test)Reject H

_{o}if*R*≥*k*_{2}, where*P*(*R*≥*k*_{2}) is obtained from the tables (One tailed test).**Large-Sample Runs Test (***n*_{1}> 10 and*n*_{2}> 10)H

_{o}: The sequence of elements has been produced in a random manner.H

_{a}: The elements have been produced in a non-random sequence.Reject H

_{o}if*z*≥ z_{α/2}or if*z*≤ –*z*_{α/2}(Two-tailed test)

Reject H

_{o}if*z*≥*z*_{α}or*z*≤ –*z*_{α}(One-tailed test)

**Chi-Squared test of independence**H

_{o}: row and column variables are independentH

_{a}: row and column variables are associated (dependent)Test statistic :

*χ*^{2}= Σ(Obs – Exp)^{2}/Exp, where the sum if taken over all cells of the table.df = (R – 1).(C – 1)

Reject H

_{o}if*χ*^{2}>*χ*^{2}_{α}.