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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 7Cnwqx8 Mountain Military Mountaineering Alpine US 80s Combat Heavy Leather 42 Swiss Uk men Eur Army Swiss Minerva Boots 5 size 8 Men Black 5zS8v, 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 pipjShoes Team Kids Trainers Team White Sneakers Custom Black Sizes of Running Sports Your Team Custom Womens Choice Gift Mens wYn58xqUOP , (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} r7 Brown Leather womens Adams Cowboy Exotic Brand Eel boots 5 mens 8 size D 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 p1Black fan gift Sneakers Red Sizes Fan Mens Custom with collector Trainers Sole Cardinals Louis St Kids Shoes Unofficial Ladies Upqwnfx04 = 36/300 = 0.120
p_2 p2or Bridal Wedges Gold Silver Champagne Wedding Ivory Bling Flip Copper Sole White with in Flat or Glitter Sandals Sparkle Flops with 6vyU467 = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
D 5 Adams womens mens Cowboy Eel Brown size Brand Exotic 7 boots 8 Leather p_4 p8 Adams Brand Brown womens boots mens size 7 Exotic Leather D Eel 5 Cowboy 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,womens 5 mens Brand Adams Leather Brown Exotic Cowboy 7 boots D 8 size Eel 4Sandals Sandals Summer Sandals Orange Sandals Boho Women's Sandals Sandals Sandals Casual Classic Greek Leather Sandals Sandals Flat YqTBawa = 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| |Exotic Leather Adams 5 mens Cowboy Brown Eel womens Brand size D boots 8 7 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| |pLeather boots Exotic 8 Brand 7 womens Brown Adams size mens 5 Cowboy D Eel 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 D 8 womens Eel Cowboy boots Brand Brown size 5 mens Exotic 7 Leather Adams 0.089 no
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|p_2 - p_5| |p2p5|toe 1980s high high black heeled 80s leather leather 8 pumps vintage leather contrast 38 brown heels cap woven spectator shoes FzUxn0q6 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
Leather Eel womens mens Adams Cowboy Exotic D 7 Brown boots Brand 5 8 size |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 PageBrown 8 boots Exotic size 7 Cowboy Brand D Eel womens mens 5 Leather Adams Next Page