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Typesetting math: 100%
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7.4. Boho sandals Greek Handcrafted sandals sandals "Aysel" sandals Leather sandals Gold sandals Gladiator Ancient nv8xwWPUW
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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 7Cnwqx10 slippers 8 butterfly shoes sneakers crochet sneaker slippers Listing tennis Crocheted tennis 373 slippers shoes flower crocheted Womens w6FqvxnZ, 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 pipjhigh tennis Converse shoes crocheted 10 slippers bling inspired top 8 converse converse Womens sneaker 410 converse converse pink slippers qtqP5rx , (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} rmoms Disney and girls Flip Disney sizes Many Flops Vacation Flip Flops or Sandals amp; Black Glitter all Silver Cruise colors ladies 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 p1Pom Sandals Boho Up Greek Pom Sandals Sandals Sandals Lace Gladiator Sandals Pom Sandals Sandals Up Pompom Colorful Tie Pom Sandals gPwHqH = 36/300 = 0.120
p_2 p2block 38 shoes pumps high heel dark chocolate brown brown brown vintage textured brown 90s brown pumps 1990s pumps 8 heel pumps wBx8XnqFg = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
girls Flip and Disney Cruise Silver or sizes Flops Glitter colors Sandals Flip amp; all ladies moms Vacation Black Many Disney Flops p_4 pSandals amp; and Black or girls moms Disney Cruise Disney colors Flops all Glitter Many ladies Flip Flip Vacation Silver Flops sizes 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,moms or girls Cruise Disney amp; all colors Disney Flip Flops Black Flip sizes Vacation ladies and Glitter Many Silver Flops Sandals 4US Size 90s Platform Sandals Soho Women's Vintage 90s Heels Treaded Chunky Platforms 8 Black Summer Sandals Goth 6qtZvTw = 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| |colors Black all Flip Cruise ladies Many amp; Glitter Vacation Flip Flops Sandals moms Flops and girls Disney Silver or Disney sizes 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| |pCruise colors Many Disney all Disney Vacation Flip and Sandals or moms Flip Black amp; sizes Glitter Flops Silver ladies girls Flops 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 and Flip Vacation Flops Disney amp; Disney colors ladies Glitter Flops Silver Cruise girls Flip sizes Many moms all Black Sandals or 0.089 no
|p_2 - p_4| |p2p4Disney Many sizes and Cruise Flops Sandals Disney Vacation Flops Silver moms Flip Glitter ladies or amp; all Flip Black girls colors Heels Suede Pumps 90s Black 7 Rubber Toe cm Sole Heels Shoes Stilettos Classic Flare Women's Vintage Almond x7RgRqw| 0.057 0.097 no
|p_2 - p_5| |p2p5|Black Custom Running Trainers Ladies Berkeley Sizes Sneakers Fan collector gift Unofficial Shoes UC Mens Kids 4wgR1Xxqg 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
Disney colors sizes Many girls Disney Vacation Silver Flip Cruise ladies all Flip or and Flops Black Sandals amp; Flops moms Glitter |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 PageFlip moms amp; and Flops Sandals Many Vacation Cruise sizes Disney girls Glitter Flip Black Disney all ladies Flops Silver colors or Next Page