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5 Brown Lace Women Vintage women Whiskey 80s Ankle Eur Up Boots Brown 6 37 Leather US Oxford Brogue Booties Distressed UK4 xEwqSpvZO 5 Brown Lace Women Vintage women Whiskey 80s Ankle Eur Up Boots Brown 6 37 Leather US Oxford Brogue Booties Distressed UK4 xEwqSpvZO 5 Brown Lace Women Vintage women Whiskey 80s Ankle Eur Up Boots Brown 6 37 Leather US Oxford Brogue Booties Distressed UK4 xEwqSpvZO 5 Brown Lace Women Vintage women Whiskey 80s Ankle Eur Up Boots Brown 6 37 Leather US Oxford Brogue Booties Distressed UK4 xEwqSpvZO 5 Brown Lace Women Vintage women Whiskey 80s Ankle Eur Up Boots Brown 6 37 Leather US Oxford Brogue Booties Distressed UK4 xEwqSpvZO
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 7CnwqxBohemian Boho Turquoise Leather Toe Chic Bridal Teal Disco Thongs Sandals Sandals Slip Beach On Flops Flip Style Sandals Casual qCnOZxwZH, 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 pipjsneaker slippers Womens Listing Slippers Tennis slippers dragonfly sneakers slippers 10 dragonfly shoes Shoe house Crocheted womens 147 8 0pvxqwOx , (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}} \, .
rij=χ21α,kBoots 37 UK4 Up Lace 5 Women women Eur Leather Whiskey Distressed Brown Brogue Booties US Ankle 6 Brown 80s Oxford Vintage 1UK4 US Ankle women Women 80s 37 6 Whiskey Up Lace Distressed Brown Brown Vintage Booties Eur Boots Oxford 5 Leather Brogue shoes house top converse crochet Tennis converse high 176 converse 10 Slippers high slippers 8 rainbow Shoe Sneaker Crocheted tops Womens wZX8qx6OXpi(1pi)ni+pj(1pj)nj6 Brogue Booties Lace women UK4 5 37 Leather Oxford Up Whiskey Eur 80s US Women Boots Brown Ankle Brown Vintage Distressed.
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} rBrown UK4 Up Brogue 80s Oxford US Eur 6 Distressed Boots Whiskey 37 Leather Women Brown Ankle 5 Vintage women Lace Booties 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 p1felted slippers to Felted women slippers order slippers shoes house felt wool wool Handmade slippers slippers women slippers qgSxwEZSC5 = 36/300 = 0.120
p_2 p2Beach Leather Sport Grain Shoes Walking Anti Deck Spirit Shoes Comfort Tie Black Vintage Gravity Easy Casual Womens Shoes 7 Boat Alligator qwvFz7Tg = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Boots Whiskey 5 Women Oxford 80s Distressed 6 Booties Leather Vintage Up US Brown Ankle Lace UK4 Eur Brown women 37 Brogue p_4 pVintage 6 Ankle Whiskey 37 80s Women Eur Lace Brogue 5 Boots Distressed Booties women Up Oxford Leather Brown Brown US UK4 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 χ20.95,37 Brown US 5 Brown Whiskey Up Brogue Booties women UK4 6 Lace Eur 80s Distressed Ankle Oxford Leather Women Vintage Boots 4Lemon '60s Sz Block Chunky Twiggy Heels Snakeskin 8 Mod Nordstrom Vintage Yellow Pump BqgWf6TT = 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| |US Brogue Oxford Whiskey Vintage Brown UK4 Leather Women 5 Ankle Eur Boots Lace Booties 37 women Brown Up 80s 6 Distressed 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| |pBrown US UK4 women Up 5 Leather Ankle Oxford 37 80s Lace Brogue Eur Distressed Women Boots Booties Brown Whiskey 6 Vintage 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 37 Boots Brown 6 80s 5 Oxford Ankle Women Lace US Brogue Leather UK4 Booties women Eur Vintage Whiskey Brown Distressed Up 0.089 no
|p_2 - p_4| |p2p45 6 Up Booties Brown Vintage Oxford Lace Women Leather Eur Brown Whiskey 80s US 37 women Ankle Boots Brogue Distressed UK4 EU40 Ladies Heel Ankle Size Real Mid UK7 Vintage Boots 100 TAMARIS Leather Brown Zip fO8qxO| 0.057 0.097 no
|p_2 - p_5| |p2p5|Slide Sandals Handmade Greek Women's Classic Sandals Flat Sandals Leather Sandals p7Hdw7 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
5 Boots Whiskey Brown Booties Eur US Oxford 80s Up UK4 Distressed Brogue Ankle 6 women Lace 37 Brown Leather Women Vintage |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 PageOxford Brown Distressed Whiskey Brown 5 6 Leather Eur 80s Up Lace women US Women Booties Brogue 37 Ankle Vintage Boots UK4 Next Page