Grgr

Section 3:

Wilconxon Rank Sum Test (2 independent samples)

Wilcoxon Rank Sum Test for Comparing Two Independent Samples

• Compares two independent samples and is a nonparametric counterpart to the two sample pooled t-test. • Test two independent population medians • Assumptions: – The samples are random and independent drawn. – The scale of measurement of the data must be at least ordinal. – Populations have approximately the same shape.

Wilcoxon Rank-Sum Test: Small Samples

• Can use when both n1 , n2 ≤ 10 • Assign ranks to the combined n1 + n2 sample observations – If unequal sample sizes, let n1 refer to smallersized sample – Smallest value rank = 1, largest value rank = n1 + n2 – Assign average rank for ties • Sum the ranks for each sample: W1 and W2 • Obtain test statistic, W1 (from smaller sample)

Wilcoxon Rank-Sum Test: Hypothesis and Decision Rule

M1 = median of population 1; M2 = median of population 2 Test statistic = T1 (Sum of ranks from smaller sample) Two-Tail Test H0: M1 = M2 H1: M1 ≠ M2

Reject Do Not Reject Reject

Left-Tail Test H0: M1 ≥ M2 H1: M1 < M2

Reject Do Not Reject

Right-Tail Test H0: M1 ≤ M2 H1: M1 > M2

Do Not Reject Reject

W1L

W1U

W1L

Reject H0 if W1 < W1L

W1U

Reject H0 if W1 > W1U

Reject H0 if W1 < W1L or if W1 > W1U

Which one provide better services? Test if the M median waiting time is shorter than the K median waiting time at 0.05 level of significance.

M: 4.21 K: 5.35 3.80 4.01 5.25 3.87 4.82 4.70 5.05 6.0 5.13

Rank

3.8 3.8 4.01 4.21 4.7 4.82 5.05 5.13 5.25 5.35 6.0 Rank sums 9 6 7 4 1.5

Rank

1.5 3 5

8 10 11 38.5

27.5

Checking the Rankings

• The sum of the rankings must satisfy the formula below • Can use this to verify the sums W1 and W2

n(n + 1) W1 +W2 = 2

where n = n1 + n2

1.Ho : M1 ≥ M2 2.H1 : M1 < M2 3.α = 0.05, n1 = 5, n2 = 6, lower critical value = 19 4.Reject Ho, if the test statistics < 19. Otherwise, do not reject Ho. 5.Test statistics = 27.5...