Appendix C—Formulae for Hypothesis Testing – Management Research Methodology: Integration of Principles, Methods and Techniques

APPENDIX C

Formulae for Hypothesis Testing

  1. Large sample statistical test

    θ may equal µ p, (µ1µ2), or (p1p2)

     

    Ho : θ = θo

     

    Ha : θ > θo or Ha : θ < θo

    (One-tailed test)

    Ha : θθo

    (Two-tailed test)

    Reject Ho if Z > Zα or if Z < – Zα

    (One-tailed test)

    Reject Ho if Z > Za/2 or if Z < –Za/2

    (Two-tailed test)

  2. Small-sample test for a population mean
     

    Ho : µ = µo

    Ha : µ > µo or µ < µo

    Reject Ho if t > tα or if t < –tα

    (One-tailed test)

    Reject Ho if t > tα/2 or if t < –tα/2

    (Two-tailed test)

  3. Small-sample test for the difference between two means based on independent random samples
     

    Ho : µ1µo = Do

    Ha : One or two-tailed hypothesis determined by the experimenter.

    Reject Ho if t > tα or if t < – tα

    (One-tailed test)

    Reject Ho when t > tα/2 or if t < –tα/2

    (Two-tailed test)

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

     

    Ho : µ1µ2 = µd = Do

    n is the number of paired differences and

    Standard deviation of paired differences.

    Alternative hypothesis Ha and α are specified by the experimenter.

  5. Test of an hypothesis about a population variance
     

    Ho : σ2 = σ2o

     

    Ha : σ2σ2o

    (Two-tailed test)

    Ha : σ2 > σ2o or Ha : σ2 < σ2o

    (One-tailed test)

    df = (n − 1)

    Reject Ho if X2 < X21–α/2 with (n – 1) df

    (Two-tailed test)

    Reject Ha : σ2 > σ2o or σ2 < σ2o

    (One-tailed test)

  6. Test of hypothesis about equality of two population variances
     

    Ho : σ21 = σ22

     

    Ha : σ21σ22

    (Two-tailed test)

    Test statistic : F = S21/S22

     
    df = (n1 –1) and (n2 – 1)
     

    Reject Ho if F > Fα/2

    (Two-tailed test)

  7. Test of a hypothesis concerning the slope of a line
     

    Ho : β1 = β1o

    Ha : Specified by the experimenter

     

    df = (n – 2)

    Reject Ho, if t < te.

  8. The Sign Test (n < 25)

    Ho: The two population distributions are identical and P(A exceeds B for a given pair) = p = .5.

    Ha: The two population distributions are not identical and p ≠ .5, or

    Ha: 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 Ho if XXL or XXU

    Where XL and XU are the lower and upper-tailed values of a binomial distribution, for a two-tailed test.

     

    Reject Ho if XXU

    Where XU is the upper-tailed value of a binomial distribution, for one-tailed test.

  9. Sign Test for large samples (n ≥ 25)
     

    Ho : p = .5

     

    Ha : p ≠ .5

    (Two-tailed test)

    Ha = 0.5 or Ha < 0.5

    (One-tailed test)

     

    Reject Ho if zzα/2 or zzα/2

     
  10. Mann–Whitney U Test

    Ho: The population relative frequency distributions for A and B are identical.

    Ha: The two population relative frequency distributions are shifted with respect to their relative locations (a two-tailed test), or

    Ha: 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 UA and UB (two-tailed test).

     

    UA = n1 n2 + n1 (n1 + 1)/2 – TA, and
    UB = n1 n2 + n2 (n2 + 1)0/2 – TB

    Where TA and TB are the rank sums for samples A and B, respectively.

    Use UA for a one-tailed test.

    Reject Ho if UUo,

    Where P(UUo) = α/2. (two tailed test).

    For a one-tailed test and a given value of α, reject Ho if UAUo = α.

  11. Mann–Whitney U Test for large samples (n1 > 10 and n2 > 10)

    Ho: The population relative frequency distribution for A and B are identical.

    Ha: The two population relative frequency distributions are not identical two-tailed test

    Ha: 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 = UA.

    Reject Ho if z ≥ zα/2 or z ≤ zα/2

    (Two-tailed test)

    Reject Ho when z < -zβ

    (One-tailed test)

  12. Kruskal–Wallis test for comparing t, more than 2 population distributions

    Ho : Distributions of all populations are identical.

    Ha : At least two of the t relative frequency distributions differ.

    Reject Ho if H > X2α.

  13. Calculation of the test statistic and the Wilcoxon Signed-Rank Test
    • Calculate the differences (XAXB) for each of the 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.
  14. Wilcoxon Signed-Rank for a paired experiment

    Ho: The two population relative frequency distributions are identical.

    Ha: The two population relative frequency distributions are not identical, in a two-tailed test

    Ha: 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 Ho if TTo, where To is the critical value in tables.

  15. Large-sample Wilcoxon Signed-Rank test for a paired experiment (n ≥ 25)

    Ho: The two population relative frequency distributions for A and B are identical.

    Ha: The two population relative frequency distributions differ in location, in a two-tailed test

    Ha: 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 Ho if zzα/2 or z ≤ –zα/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 Ho when z > zα.

    To detect a shift in the opposite direction, let T=T and reject Ho if z < – zα (One-tailed test).

  16. Friedman Test for randomised block designs

    Ho: The probability distributions for the t treatments (populations) are identical.

    Ha: At least two of the t treatments have different probability distributions.

    Reject Ho if χ2 > χ2α.

  17. Runs Test

    Ho: The sequence of elements call them S’s and F’s, has been produced in a random manner.

    Ha: The elements have been produced in a non-random sequence (a two-tailed test)

    Ha: 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 Ho if Rk1 or Rk2

    Where P(Rk1) +P(Rk2) = α and k1 and k2 are obtained from the tables (Two-tailed test).

    Reject Ho if Rk1, where P(Rk1) = α and k1is obtained from the tables (Two-tailed test)

    Reject Ho if Rk2, where P(Rk2) is obtained from the tables (One tailed test).

  18. Large-Sample Runs Test (n1 > 10 and n2 > 10)

    Ho: The sequence of elements has been produced in a random manner.

    Ha: The elements have been produced in a non-random sequence.

    Reject Ho if z ≥ zα/2 or if z ≤ – zα/2

    (Two-tailed test)

    Reject Ho if zzα or z ≤ –zα

    (One-tailed test)

  19. Chi-Squared test of independence

    Ho: row and column variables are independent

    Ha: row and column variables are associated (dependent)

    Test statistic :

    χ2 = Σ(Obs – Exp)2/Exp, where the sum if taken over all cells of the table.

     

    df = (R – 1).(C – 1)

    Reject Ho if χ2 > χ2α.