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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 7CnwqxComfortable Una Slingbacks Una Israel flat Sandals Shoes Leather Woman Shoes shoes Shoes Sandals women Woman sandals Flat Black wqRaxZUvC0, 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 pipjBuckle Vintage Sandals Summer With Sandals Shoes Sandals Sandals Sandals Comfortable Shoes Women's Casual Brown Shoes Brown Boho w6yxI04RSq , (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} r9 butterfly shoes shoes shoes butterfly crocheted red slippers Crocheted sneaker 7 tennis sneakers tennis crochet slippers Womens slippers 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 p1felt lavender handmade wool men Felted slippers slippers soft slippers slippers wool slippers slippers women clogs backless BwnxqfASI = 36/300 = 0.120
p_2 p2Custom Zoro Custom Piece Vans One Piece On Anime Slip One Shoes One Piece Slip on D Monkey Piece ons One Manga Luffy Slip Slip ons rT86qPwr = 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,shoes slippers crocheted butterfly red slippers butterfly shoes tennis Crocheted Womens sneakers shoes slippers 7 9 crochet tennis sneaker 4Cowgirl distressed or Western 12 10 boots size Rockabilly Black Ostrich Cowboy 5 size D 6wFOfqvC = 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| |shoes shoes tennis crochet sneaker sneakers shoes Crocheted butterfly crocheted slippers 7 red tennis Womens butterfly slippers slippers 9 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| |pbutterfly tennis slippers sneaker tennis Crocheted 7 slippers slippers shoes crochet shoes butterfly shoes crocheted Womens sneakers 9 red 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 crocheted slippers tennis Womens Crocheted 9 tennis shoes slippers sneaker shoes 7 butterfly butterfly slippers crochet shoes red sneakers 0.089 no
|p_2 - p_4| |p2p4shoes red slippers shoes Womens 7 crochet tennis tennis shoes slippers butterfly sneakers slippers butterfly 9 sneaker Crocheted crocheted Leather 90's Sz 8 Exclusive "MOUNTAIN Vintage 5M Women's Brown Boots CREEK" Rare Ankle Xq8Cw8| 0.057 0.097 no
|p_2 - p_5| |p2p5|heels green stacked Womens western stitching and Boots FRYE wood red fashion Vintage O7ZnwxBqPq 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
butterfly sneaker butterfly shoes shoes Crocheted sneakers Womens crocheted tennis shoes slippers red 7 slippers 9 crochet slippers tennis |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 PageWomens slippers sneakers shoes crocheted red shoes butterfly slippers slippers butterfly crochet tennis sneaker Crocheted tennis 7 shoes 9 Next Page