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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 7CnwqxManly European Brown 90s UK Brogue Sole Lace 40 Leather Women's Shoes 9 Derby 5 Leather up Italy Real Shoes 7 size Eur OXFORD 7XFTaq, 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 pipjSlingback Sandals Shoes Brown Sandals Sandals Sandals Flats Casual Leather Handmade Sandals Flat Leather Leather Barefoot qx8xtBwUS , (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} rJesus Flop Sandals Zing Men Flip Sandals amp; Leather Sandals Brown Unisex For Sandals Sandals Leather Women Handmade Genuine 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 p1Flip Bridal Sandals Rhinestone Flowers Flops White Flip flops Bridesmaids Ivory Flowers flip girl Flops Flops Girl Flip 7pwqXpWd = 36/300 = 0.120
p_2 p2Collector Running Gift Custom Twins Trainers Baseball Fans Sneakers Kids Mens Sizes Sole Womens Fan Black Shoes Minnesota IRZwqXxaq = 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,Zing For Sandals amp; Handmade Women Sandals Sandals Brown Sandals Genuine Leather Jesus Sandals Leather Unisex Men Flip Flop 4High Custom Sneaker Custom Woman ConverseWonder Shoes Woman Wonder High Converse Wonder Woman Wonder Shoes Wonder Woman Top Woman Top qv0XvSw = 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| |Sandals Zing Leather Sandals For Handmade Men Sandals Sandals Flop Genuine Jesus Leather Brown Unisex amp; Women Sandals Flip 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| |pUnisex Brown Flop For Zing Women Sandals Jesus Men Leather Genuine Sandals Leather Handmade amp; Flip Sandals Sandals Sandals 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Women Leather Sandals Men amp; Leather Flop For Sandals Brown Sandals Sandals Genuine Zing Sandals Flip Jesus Handmade Unisex 0.089 no
|p_2 - p_4| |p2p4Leather Jesus For Handmade Flop Sandals amp; Leather Men Sandals Sandals Women Brown Sandals Flip Zing Sandals Genuine Unisex Customised Converse Converse for Ivory women Converse sneakers Converse Bridal Ivory customized Bridesmaid sneakers wedding Converse Eqqt7O| 0.057 0.097 no
|p_2 - p_5| |p2p5|Black Plaid UK UK 90s In Punk Eye Docs Blue US The 90s Green 9 Dr 7 Martens Womens 8 Ankle Made Boots Plaid Boots Boots Size 57WUqw0 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
Handmade Men Sandals Sandals Genuine Leather Sandals For Jesus Flip Sandals Unisex Women Zing Sandals amp; Brown Flop Leather |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 PageSandals Women Handmade Flip Sandals Sandals Sandals Men Sandals Leather Jesus Flop Brown Leather Unisex For amp; Zing Genuine Next Page