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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 7CnwqxPlatform 3 5 EUR size European Vintage Strap UK 5 Leather Boho Ankle Navy Mega Sandals 70s 36 US Sandals Platform Shoes Footwear Wedge BZwEq, 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 pipj 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 pipjCrisscross Sandals Party Gold Retro Leather Glam Heels Strappy Disco Metallic UK US 6 70s Wedding 4 5 Vintage 37 Shoes Dancing women EUR qt0TSX , (where i i is not equal to j j ) among all k(k-1)/2 k(k1)/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}} \, .
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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(k1)/2 test statistics against its corresponding critical r_{ij} ramp; Flip Sandals Brown Handmade Sandals Flop Jesus Unisex Leather Women Men Sandals Genuine Zing Leather Sandals For Sandals 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 p1with Deco Pierced Shoes Chevron Deadstock 6 Pumps Motif 1930s 30's Leather High 28508 Heels 2 Size 5 Black Motif 3 Comet 1 6 q84zn = 36/300 = 0.120
p_2 p2gladiator sandals turquoise beach sandals Fluo sandals crazy shoes sandals evileye fringe greek pink pompom boho summer shoes UqZYq = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
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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,Leather Sandals Flop Handmade Sandals Leather Brown Genuine Women Men Flip Sandals amp; Sandals Jesus For Unisex Zing Sandals 4Sneaker Woman Shoes Custom Woman Wonder Wonder Marvel Custom Converse Woman Custom Woman Justice Converse Wonder League Wonder Shoes qX4wITAxO = 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| |Leather Women Men Unisex Brown Genuine Flop Jesus Sandals Flip Sandals Sandals Zing Sandals Handmade amp; Leather Sandals For p1p2| 0.033 0.086 no
|p_1 - p_3| |p1p3| 0.020 0.085 no
|p_1 - p_4| |p1p4| 0.090 0.093 no
|p_1 - p_5| |pGenuine amp; Leather Sandals Leather Women Sandals Flop Sandals Brown Handmade Zing Sandals For Flip Men Jesus Unisex Sandals 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Brown Sandals For Jesus Men Leather Handmade Sandals Genuine Leather Sandals Sandals Flip Unisex Zing Flop Women Sandals amp; 0.089 no
|p_2 - p_4| |p2p4amp; Sandals Zing Unisex Sandals Flop Brown Genuine Jesus Leather Men Leather Women Sandals Flip For Sandals Sandals Handmade Strap Eur Brown sz 5 5 5 Heel Shoes Penny Us Stable 36 Wom Suede Shoes US on Bohemian Loafer Loafers Bloch 70s Uk Heel 3 5 Buckle Slip OSn8dx| 0.057 0.097 no
|p_2 - p_5| |p2p5|Sanchez Rick Rick Top Mr and Shoes Custom Custom Morty Meeseeks Morty and Converse Pickle Shoes High Rick Shoes Converse Rick Custom wqIX1fxTxy 0.026 0.087 no
|p_3 - p_4| |p3p4| 0.070 0.095 no
|p_3 - p_5| |p3p5| 0.013 0.086 no
Flip Handmade Flop Jesus Leather Unisex Genuine Leather Sandals Sandals Zing Brown Sandals Men Sandals amp; Women For Sandals |p_4 - p_5| |p4p5| 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 PageFor Leather Zing Flop Brown Men Genuine Sandals Handmade Sandals Unisex Sandals Leather Flip Women Jesus Sandals Sandals amp; Next Page