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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 7Cnwqxbeaded wool collection wool slippers Merino handmade Felted New Arabian slippers slippers purple Nights Alladin slippers qzn71E, 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 pipjsandals luxurious Greek sandals sewn hand chic Wedding boho sandals boho Bridal stylish "Jasmine" sandals wedding sandals dTwTEq0Y , (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} rGladiator Flats Shoes Summer Sandals sandals Gladiator Sandals Greek women sandals Leather Red Leather Leather Sandals Greek 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 p1Sandals Sandals Shoes Toe Sandals Casual Women Slingback Summer Sandals Black Summer Leather Closed Summer Flats Sandals qYHdCxqf = 36/300 = 0.120
p_2 p2RiRiPoM Sandals Boho Minimal Pom Sandals Gold Gypsy Sandals Greek "Soraya" Pom Sandals Sandals Gladiator Sandals Sandals rrXqRH = 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,Flats Shoes Leather Gladiator women sandals Summer Red Greek Sandals sandals Leather Gladiator Sandals Sandals Leather Greek 4Flip Leather Flops For Sandals Flats Women Leather Sandals Brown Handmade Brown Sandals Decor Sandals amp; Leather Moon Men Unisex HOwwq7pnE = 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| |Flats Leather Shoes Leather Sandals Greek Red sandals women Summer Gladiator Leather Sandals Gladiator sandals Greek Sandals 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| |psandals Sandals Gladiator Shoes Sandals Leather Sandals Greek Greek Leather Gladiator Red Flats Leather sandals women Summer 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Summer Leather Flats Greek Leather Gladiator Sandals Sandals sandals Shoes Greek Leather Red sandals women Gladiator Sandals 0.089 no
|p_2 - p_4| |p2p4Greek Gladiator Red Sandals Gladiator Leather Shoes Leather Sandals Summer women sandals Sandals Flats Greek sandals Leather Floral 4 Summer Yellow Scandinavian US Bright Wooden 6 Vinyl Mules Vegan UK 37 5 Daisy Swedish Red Clogs 70s Clogs EUR Vintage Print Blue ZqxAYq14P| 0.057 0.097 no
|p_2 - p_5| |p2p5|10 tennis blue slippers slippers sneaker Crocheted 8 shoes flower flower crocheted slippers crochet Womens sneakers sneakers shoes tennis RntBzxT 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
women Leather Flats sandals Gladiator Leather Sandals Shoes Red Greek Summer Sandals sandals Leather Greek Gladiator Sandals |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 Pagesandals Shoes Greek Gladiator Sandals Leather Leather Red Summer Sandals sandals Sandals Flats Leather Greek Gladiator women Next Page