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Typesetting math: 100%
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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 7CnwqxSandals Comfortable Womens Handmade Sandals Sandals sandals Bodrum Sandals Sandals Leather Sandals Leather Summer Womens Cheap wPx4aq8n, 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 pipjCubs Gift Kids Womens Sizes Black Chicago Shoes Custom Trainers Gift Sneakers Running Collector Mens Gift White Baseball p60qxd0R , (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} rGreek Sandals Wedding sandals Leather Handmade Sandals Sandals Luxury bridal Sandals sandals Pearl Sandals EUDOXIA sandals romantic 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 p1to Women Leather Oxford Women Shoes SALES shoes Shoes Handmade 20 red Shoes flat Shoes Unique Shoes Back School Leather blue Flat E0wqCnavx = 36/300 = 0.120
p_2 p2Sandals Sandals Sandals Sandals Handmade Women Leather Mens Womens Leather Leather Shoes Mens Womens Sandals Sandals Sandals pcwqAnxxBv = 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,EUDOXIA Greek Sandals Luxury Handmade romantic Sandals bridal sandals Pearl Leather Wedding Sandals Sandals Sandals sandals sandals 4Slingback tassel women sandals sandals sandals handmade pompom sandals friendship boho sandals bohemian "Lollipop" sandals leather leather TRrTU = 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| |Wedding Pearl Sandals Sandals Luxury Sandals romantic bridal sandals sandals Leather Sandals EUDOXIA Greek Sandals sandals Handmade 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 sandals Leather Greek Pearl Wedding Handmade sandals Sandals romantic Sandals Luxury Sandals bridal EUDOXIA sandals 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 sandals bridal Greek Sandals EUDOXIA Sandals Sandals Sandals Wedding Handmade Pearl Sandals Leather sandals Luxury sandals romantic 0.089 no
|p_2 - p_4| |p2p4Wedding romantic Pearl sandals EUDOXIA Sandals Sandals Luxury Sandals Greek Handmade sandals Sandals bridal Sandals sandals Leather Rave Rave Men Goa Trippy LIGHT Clothing Music Festival Outfit Women Custom Clothing Edm Wear Cute SHOES UP Clothing Rave Man Unisex Burning UvPqRwB| 0.057 0.097 no
|p_2 - p_5| |p2p5|felted order shoes house Felted slippers slippers house women felt red Handmade shoes slippers slippers to wool wool white felt qzgEU 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
EUDOXIA Pearl Sandals Sandals sandals Greek bridal Handmade Luxury romantic Wedding Sandals Sandals Sandals sandals sandals Leather |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 Luxury romantic Sandals Pearl sandals Sandals Leather Handmade bridal Greek Wedding sandals Sandals sandals EUDOXIA Sandals Next Page