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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 7CnwqxChampagne in 25 Glitter inch Platform Flops Ivory Wedge inch 1 Flip and heel with 3 amp; Flower 2 low 5 Vintage medium or Soles White fPwOqFE4, 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 pipjGray with Flip Ivory Glitter Flops Emerald Cur 3 Wedding inch Wedge and the Painted Sides with for Silver 5 Bride Pewter Embellishment qSg0nztn , (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} rJesus Unisex Flower Brown cross For Sandals Sandals Genuine Flop Sandals Flip amp; Sandals Sandals Men Handmade Women Leather Leather 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 p1Wedding last any the Flops White Platinum with Bridal Platform Flip Ivory Flops Flip personalized new Silver or colors mrs name in Wedge qZnS0pwn = 36/300 = 0.120
p_2 p2felt unisex wool slippers grey shoe men slippers felted Wool felted slipper natural felt women clogs wool clogs slippers felt wool P1Tqw1 = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Sandals Leather Genuine Sandals cross Women Flop Sandals Brown Flip Flower Unisex Sandals Leather Jesus Handmade amp; Men For Sandals p_4 pSandals cross Brown Flip Unisex Handmade Genuine Women Sandals Men Sandals Sandals Sandals Leather For Leather Flop Flower amp; Jesus 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,Sandals Unisex Handmade Flop Men Sandals Flip Brown amp; Flower Genuine Sandals Leather cross Jesus Sandals Leather Sandals Women For 4~ Bride Heels Sweet Wedding Prom Party 16 the Graduation Pearl of Ivory High ~ Christening Mother Bridesmaid Shoes Pageant wpXBqAB = 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| |Brown Flower Jesus Handmade Sandals Flip Sandals Leather Flop Genuine Sandals Sandals Unisex Men For cross Leather Sandals amp; Women 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| |pLeather Handmade Flip Sandals Brown Leather amp; Men Sandals Genuine Jesus Flop Sandals Unisex Flower Sandals Sandals For Women cross 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Brown Women Genuine Jesus Leather Flower amp; cross Sandals Leather Flop Sandals Flip Sandals Handmade Sandals Unisex Sandals For Men 0.089 no
|p_2 - p_4| |p2p4amp; cross Men Jesus Flip Women Flower Leather Handmade Sandals Genuine Sandals Sandals Brown Unisex Sandals Flop Leather For Sandals Shoes Shoes Shoes Every Shoes Wooden Comfortable Flat Shoes Leather Textured Heels Leather Shelly Day Shoes Black Women gHX1g45q| 0.057 0.097 no
|p_2 - p_5| |p2p5|Sandals Sandals Leather amp; Jesus Sandals Men Sandals For Flop Sandals Unisex Genuine Triple Flip Women Handmade Red Leather pT7qgBg 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 Flip Leather Sandals amp; Genuine cross Flop Sandals Sandals For Flower Sandals Unisex Sandals Brown Leather Jesus Handmade Men |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 PageBrown Genuine For Sandals Leather Flower cross Sandals Men Jesus Flip Sandals Handmade Leather Unisex Women Flop amp; Sandals Sandals Next Page