Test at the = .05 level: This is a two-tailed test and n = 8, so find W L and W U in appendix P: W L = 3 and W U = 33 The calculated test statistic is W = R+ = 27
W L = 3 and W U = 33 W L < W < W U so do not reject H 0 (there is not sufficient evidence to conclude that the median class size is different than 40) (continued) W L = 3 do not reject H 0 reject H 0 W = R+ = 27 W U = 33 reject H 0
The W test statistic approaches a normal distribution as n increases
For n > 20, W can be approximated by
where W = sum of the R+ ranks d = number of non-zero d i values
15.
Nonparametric Tests for Two Population Centers
Nonparametric
Tests for Two
Population Centers
Wilcoxon Matched-Pairs Signed Rank Test Mann-Whitney U-test Large Samples Small Samples Large Samples Small Samples
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Mann-Whitney U-Test Used to compare two samples from two populations Assumptions: The two samples are independent and random The value measured is a continuous variable The measurement scale used is at least ordinal If they differ, the distributions of the two populations will differ only with respect to the central location
If the sum of rankings from one sample differs enough from the sum of rankings from the other sample, we conclude there is a difference in the population medians
Mann-Whitney U-Test (continued)
19.
Mann-Whitney U-Test (continued) Mann-Whitney U-Statistics where: n 1 and n 2 are the two sample sizes R 1 and R 2 = sum of ranks for samples 1 and 2
20.
Mann-Whitney U-Test (continued) Claim: Median class size for Math is larger than the median class size for English A random sample of 9 Math and 9 English classes is selected (samples do not have to be of equal size) Rank the combined values and then split them back into the separate samples
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Mann-Whitney U-Test H 0 : Median M ≤ Median E H A : Median M > Median E Claim: Median class size for Math is larger than the median class size for English Note: U 1 + U 2 = n 1 n 2 (continued) Math: English:
Large Sample Example Since the alternative hypothesis indicates that population 2 has a higher median, use U 2 as the test statistic Compute the U statistics: (continued)
33.
Large Sample Example Since z = -2.80 < -1.645, we reject H 0 Reject H 0 = .05 Do not reject H 0 0 (continued)
Kruskal-Wallis Example (continued) Compare H = 6.72 to the critical value from the chi-square distribution for 5 – 1 = 4 degrees of freedom and = .05: There is not sufficient evidence to reject that the population medians are all equal
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