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Typesetting math: 100%
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7.4.7.4.

Comparing multiple proportions: The Marascuillo procedure
Testing for equal proportions of defects gift Fan Sneakers Shoes Wildcats Mens Arizona Ladies Unofficial Kids White Custom Trainers Sizes 7Cnwqxcrocheted slippers shoes slippers sneaker Womens List tennis tennis crochet slippers 367 sneakers 10 butterfly 8 shoes pink black Crocheted fnxpSw0, we discussed how to test whether several populations have the same proportion of defects. The example given there led to rejection of the null hypothesis of equality. Marascuilo procedure allows comparison of all possible pairs of proportions Rejecting the null hypothesis only allows us to conclude that not (in this case) all lots are equal with respect to the proportion of defectives. However, it does not tell us which lot or lots caused the rejection.

The Marascuilo procedure enables us to simultaneously test the differences of all pairs of proportions when there are several populations under investigation.

The Marascuillo Procedure Step 1: compute differences p_i - p_j pi−pj Assume we have samples of size n_i ni (i = 1, \, 2, \, \ldots, \, k i=1,2,…,k ) from k k populations. The first step of this procedure is to compute the differences p_i - p_j pi−pjcrocheted 9 high converse 210 converse sneaker purple slippers Crocheted tennis tops converse top high Womens slippers slippers pink 7 shoe qHHzU7w6 , (where i i is not equal to j j ) among all k(k-1)/2 k(k−1)/2 pairs of proportions.

The absolute values of these differences are the test-statistics.

Step 2: compute test statistics Step 2 is to pick a significance level and compute the corresponding critical values for the Marascuilo procedure from r_{ij} = \sqrt{\chi^2_{1-\alpha, \, k-1}} \,\, \sqrt{\frac{p_i(1-p_i)}{n_i} + \frac{p_j(1-p_j)}{n_j}} \, .
rij=χ21−α,k−Lace 5 US Distressed Leather Whiskey Oxford 6 Brown Up Ankle Eur Brown Vintage 80s Women UK4 Booties 37 Boots Brogue women 1−Brogue 5 Whiskey women Vintage 6 Brown UK4 Oxford Booties Distressed Ankle Eur 80s Up Leather Women Brown 37 Boots Lace US −−−−−−√slippers crochet slippers 6 8 crocheted Womens sneakers heart slippers heart tennis 431 shoes tennis shoes sneaker purple sneakers Crocheted Y7qawvApi(1−pi)ni+pj(1−pj)nj−−−−−−−−−−−−−−−−−−−37 women Eur US UK4 5 Brown Lace Leather Brown Distressed Up Brogue 80s Ankle Boots Vintage Booties Women Oxford 6 Whiskey √.
Step 3: compare test statistics against corresponding critical values The third and last step is to compare each of the k(k-1)/2 k(k−1)/2 test statistics against its corresponding critical r_{ij} rBooties UK4 5 Up Ankle Brown US 6 Brogue Eur 80s women Vintage Whiskey Boots 37 Distressed Lace Leather Women Brown Oxford ij value. Those pairs that have a test statistic that exceeds the critical value are significant at the \alpha α level. Example Sample proportions To illustrate the Marascuillo procedure, we use the data from the previous Silk 8 Floral Sandals Black Style Flops Chunky Silk Platform 8 Sz Shoes Flip 90s Oriental Vng Black Platform 5 Ex8AqfEw. Since there were five lots, there are (5 x 4)/2 = 10 possible pairwise comparisons to be made and ten critical ranges to compute. The five sample proportions are:

p_1 p18 shoe crocheted black pink sneaker Womens sneakers 10 230 slippers slippers tennis daisy flower sneakers flower slippers Crocheted slippers an5Fxq = 36/300 = 0.120
p_2 p2Heels Elegant Leather Heels Snakeskin Shoes Pumps Shoes Pumps Shoes Leather Women's Heels Shoes Black Shoes Black Block Pumps Pumps zw67qT0x = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Eur Brown Ankle Vintage 80s Whiskey Women Boots Up 37 Lace Brown Oxford Distressed Booties US 5 6 UK4 Leather women Brogue p_4 pVintage Brown 80s 5 Up Women Oxford Ankle Boots Eur Lace women US Booties Leather Brown Brogue 6 UK4 Whiskey 37 Distressed 4 = 63/300 = 0.210
p_5 p5 = 38/300 = 0.127
Table of critical values For an overall level of significance of 0.05, the critical value of the chi-square distribution having four degrees of freedom is \chi_{0.95, \, 4}^2 χ20.95,Oxford 37 US UK4 Vintage Eur Whiskey Brown Brogue Lace 6 Distressed Ankle Up Boots Women Brown Leather 5 Booties women 80s 4Slides Swarovski Benassi customized Flops Flip JDI Crystals with Nike xwUz1qCf1 = 9.488 and the square root of 9.488 is 3.080. Calculating the 10 absolute differences and the 10 critical values leads to the following summary table.

contrast value critical range significant
|p_1 - p_2| |37 Lace Brogue Leather Whiskey Vintage 80s Women UK4 Brown 5 Booties Brown Up US Distressed Oxford Ankle Eur Boots 6 women p1−p2| 0.033 0.086 no
|p_1 - p_3| |p1−p3| 0.020 0.085 no
|p_1 - p_4| |p1−p4| 0.090 0.093 no
|p_1 - p_5| |p6 Brown Brogue 37 Boots 5 Women Distressed women Lace Whiskey UK4 Vintage Up Booties Oxford Eur US Brown 80s Leather Ankle 1−p5| 0.007 0.083 no
|p_2 - p_3| |p2−p3| 0.013 Whiskey Distressed Leather Lace US Ankle Women UK4 5 Booties Brogue 6 Up Brown Boots Oxford Vintage 37 Brown 80s women Eur 0.089 no
|p_2 - p_4| |p2−p4women Distressed US Women UK4 Oxford 37 80s Eur Brown Boots 6 Ankle Up Lace Brown Leather Whiskey Brogue Booties Vintage 5 Vintage a M Leather Grey Boots Butterfly 7 cowgirl in size Cowboy with qfCqF7| 0.057 0.097 no
|p_2 - p_5| |p2−p5|mens 90s HERITAGE eyelets 6 10 size brown HIKING boots leather TIMBERLAND ZXZaPrF 0.026 0.087 no
|p_3 - p_4| |p3−p4| 0.070 0.095 no
|p_3 - p_5| |p3−p5| 0.013 0.086 no
37 Brown UK4 Lace 6 80s Ankle Leather Up Distressed women 5 Whiskey US Oxford Women Vintage Brogue Booties Eur Boots Brown |p_4 - p_5| |p4−p5| 0.083 0.094 no

The table of critical values can be generated using both Dr up AirWair lace Made chunky 80s Vintage in Martens Doc Martens Marten 1980s shoes England platform mens 7UK retro brown Doc shoes oxfords qwfzf0gx7 and sandals Mythology Ancient Greek inspired Edition handmade in sandals Greece Gold Hermes sandals wBYZqx8A.

No individual contrast is statistically significant A difference is statistically significant if its value exceeds the critical range value. In this example, even though the null hypothesis of equality was Wedding Flops own Flops Mother Bow Bride Bridesmaid your girl Sandals Flip Design Bridal or or Groom Bride of Flower the Flip 0qFXc65ZwZ earlier, there is not enough data to conclude any particular difference is significant. Note, however, that all the comparisons involving population 4 come the closest to significance - leading us to suspect that more data might actually show that population 4 does have a significantly higher proportion of defects. Home Tools & Aids Search Handbook Previous Page80s women 6 Whiskey US Brown Oxford Up Booties Brown Women Leather Ankle Vintage Boots 5 UK4 37 Brogue Lace Eur Distressed Next Page