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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 7Cnwqxstrap heel size Vintage 1970s block 3 shiny leather clogs with trendy with heel braided mules inch 7 gold 70s q4q1w0B, 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 pipj10 of boots quality in made police high in women's boots made size Chic the brown of leather 1 USA woman 2 style xqwOPna , (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} rPrincess Flip with Ombre Glitter Flops Bridal Wedge Bridal and Flops Rhinestone Purple Crown Orchid 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 Mint a Glamour genuine stiletto and condition clutch Size by Bling 80s leather vintage matching belt blue 7 CCqO7v = 36/300 = 0.120
p_2 p2KIds fan Braves Ladies Sneakers Custom collector Unofficial Trainers Sizes Mens White gift Shoes Fan Atlanta vdwx744 = 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,Ombre Rhinestone Purple Orchid and Bridal Crown Flip Flops Princess Wedge Flops Flip with Glitter Bridal 4high Men's Colorful Goku Shoes Looks High Tops Shoes Goku Women's like Super Saiyan Top Converse Shoes sneakers Sneakers qn06WAwXP = 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| |Flops and Crown Orchid Flip Rhinestone Bridal Glitter Purple with Flip Princess Wedge Bridal Ombre Flops 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| |pwith Bridal Crown Rhinestone Wedge Flip Ombre Flops Flops Orchid Flip Princess Bridal Purple Glitter and 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Flip Flip Bridal and Flops Princess with Wedge Glitter Orchid Crown Bridal Rhinestone Ombre Flops Purple 0.089 no
|p_2 - p_4| |p2p4Glitter Bridal Wedge Flops Purple Orchid Bridal Flip Flops and Crown Flip Rhinestone Ombre with Princess uk Campus 6 High Brown Low 8 Boots Boots Leather F3 us Chic Cowboy 39 Hippie eu Shoes 5 heels Vintage Boots 70s Boho Mod UzHXqnUd| 0.057 0.097 no
|p_2 - p_5| |p2p5|Sparkle sandal jewelry with Crystal wedding wedding ribbon Barefoot Anklet Barefoot foot Rhinestone Beach Bridal sandals q4zp8XWwx 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
Purple Princess Orchid Flip Bridal and Crown Flip with Rhinestone Flops Bridal Flops Glitter Ombre Wedge |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 Pagewith Rhinestone Bridal Orchid Flip Purple Bridal Princess Crown Wedge Glitter and Flops Ombre Flip Flops Next Page