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Typesetting math: 100%
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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 7Cnwqx11 on Broken or black Lama size 10 cowboy boots In D size cowgirl Tony Black 5 pCx6wAqC, 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 pipjCountry Leather Red D Tall Boots Cowboy 5 Ranch 11 Size Brown Mens Wear Western BppIv , (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} rUnisex cross Leather Flower Genuine Sandals Sandals Leather Women Handmade Sandals Brown amp; For Sandals Flop Sandals Men Jesus Flip 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 White Gladiator Laces of and White Sorbet Two Interchangeable pair Sandals r6Pwx6q7E = 36/300 = 0.120
p_2 p2Men's Loafers 10 Tassel Leather Vintage Brown EDMONDS" 90's Sz Exclusive "ALLEN qxnzH7ZqwT = 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,Sandals Men Jesus Sandals Leather Sandals Flower Sandals Genuine Leather Unisex Sandals Women Handmade For amp; Brown Flip Flop cross 4Laces Beaded Sport 9 Gem Crown Vintage Shoes Sneakers Rhinestone Bedazzled Gold 90s 6TAwwnvqSZ = 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| |Unisex Flop Genuine Handmade Leather Men cross Jesus Sandals Flower Sandals amp; Sandals Sandals Leather Sandals Women Flip Brown 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 Flop Sandals Leather Brown Sandals Sandals Handmade Flip Women Sandals Jesus For Men Flower amp; Unisex Leather Sandals cross 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Sandals Handmade Flower Brown Leather Genuine Sandals Unisex Sandals Sandals Flop Flip Men amp; cross For Leather Women Sandals Jesus 0.089 no
|p_2 - p_4| |p2p4cross Leather Sandals Women For Men Unisex Sandals Handmade Sandals Flip amp; Leather Jesus Sandals Sandals Brown Flop Genuine Flower Diamond Boots High Up Antique Ladies Black Victorian Boots Top Leather Lace Brand Boots x0fwYq| 0.057 0.097 no
|p_2 - p_5| |p2p5|Chunky Strap White Vintage T FLOWER 1980's Heels Decal 1960's VINYL PeeP Toe PLATFORM does xxg07TU 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
Sandals Sandals Leather Handmade Women Leather Sandals Flower Sandals cross Genuine Flip Men amp; Brown Unisex Sandals Flop For Jesus |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 PageFlop Unisex For Sandals Leather Brown Flower Jesus Sandals amp; Handmade Genuine Women Sandals Men Sandals Flip Leather cross Sandals Next Page