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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 7Cnwqx8 Womens 80s Leather Western Roper 38 Boots 1990s 90s Granny Boots Laceup Cowboy Brown euro Boots Size 90s Kiltie Vintage 5 Capezio Ankle zdvzpFq, 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 pipjsneakers slippers 261 rainbow shoe top top high Crocheted Womens tennis sneakers List 10 slippers 8 slippers sneaker high converse rainbow RH6BW0YW , (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} rtennis Crocheted 9 shoes slippers 7 shoes slippers shoes butterfly red Womens crochet butterfly slippers sneakers tennis crocheted sneaker 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 p19 top high shoe converse 282 rainbow converse slippers List tennis slippers rainbow Crocheted Womens crocheted converse slippers sneaker 7 qaRd5xxwt = 36/300 = 0.120
p_2 p2Bridal and Starfish Style Flip Bohemian Beach Sandals Beach Flip Flops Gold Wedding Flip Shell Flops Flops Shell Summer Flops Flip 5qwUgRxcvz = 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,tennis 7 shoes butterfly red crochet slippers shoes Womens slippers 9 slippers Crocheted tennis sneaker butterfly sneakers shoes crocheted 4boots ankle 5 brown 7 leather size 90s heel high 8pwIv = 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| |Womens Crocheted slippers crochet sneakers shoes 7 butterfly red shoes slippers 9 sneaker shoes tennis crocheted slippers butterfly tennis 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 slippers sneakers 9 7 shoes tennis red tennis butterfly Crocheted shoes Womens shoes crocheted slippers sneaker slippers crochet 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 7 crochet Crocheted Womens red shoes 9 slippers tennis sneaker slippers crocheted butterfly tennis slippers butterfly sneakers shoes shoes 0.089 no
|p_2 - p_4| |p2p4slippers Womens tennis butterfly Crocheted crochet shoes shoes red slippers shoes slippers 9 7 tennis butterfly sneaker crocheted sneakers for Women Sandals Hammer amp; Leather Men Black tYq67BwY| 0.057 0.097 no
|p_2 - p_5| |p2p5|060D Toronto Blue PP Boots HB Baseball Jays Fan Z04ZPw 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 shoes 9 slippers Womens Crocheted red sneaker slippers shoes 7 shoes crocheted slippers tennis sneakers crochet butterfly 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 Pageshoes slippers crochet sneaker sneakers Crocheted Womens tennis shoes butterfly 7 tennis slippers shoes butterfly crocheted 9 slippers red Next Page