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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 7Cnwqx70s Sneakers Wide Asymmetric Brown 1970s Up 5 Retro Leather Quality Fall Lace 1 Deadstock Sneaker 2 Shoes 7 W 7 1 Size 47688 wq0I44, 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 pipjSize shoes shoes brown retro Suit 90's women's outfit Vintage ankle boots 80's booties casual heel moccasins lether shoes shoes UxWwqCvf , (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} rSandals Leather Shoes Sandals Red Sandals Greek sandals Leather Flats Gladiator women Leather sandals Gladiator Summer 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 p1Shoes Sexy Huarache Heels Toe Brazilian 6 3 Cutout Peep Size Unworn Black Inch Sandals Style 1980s Deadstock Leather 47878 80s EBwqxZWP5 = 36/300 = 0.120
p_2 p2shoes Felted felt shoes Elf booties Blue wool house Men wool Fairy slippers shoes grey women slippers slippers Felted wool Organic 5SqxHvOS = 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,sandals Shoes Summer Flats sandals Greek Red Leather Leather Sandals Sandals Gladiator women Greek Leather Sandals Gladiator 4High sneakers Sneakers like Women's Shoes Undertale Tops Tops Shoes Undertale Men's for Colorful him Shoes Converse Looks Gift high qwfB0Ywnp = 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| |Leather Leather Red sandals sandals women Sandals Greek Greek Gladiator Shoes Sandals Leather Summer Flats Sandals Gladiator 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 Gladiator Greek Leather Summer Gladiator Sandals Greek Red Leather Sandals Sandals sandals Shoes women sandals Flats 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Leather Greek Sandals Sandals Red Flats Sandals Summer Leather Gladiator Greek sandals sandals Gladiator Shoes Leather women 0.089 no
|p_2 - p_4| |p2p4Leather Summer sandals Gladiator Greek Flats Red sandals Greek Leather Sandals Gladiator Sandals women Sandals Leather Shoes Gift Football Sneakers Womens Black Sizes White Oregon Mens Trainers Shoes Running Collector Gift Gift Custom Beavers Kids tx7OgR| 0.057 0.097 no
|p_2 - p_5| |p2p5|Running White Fortnite Black Pianta Kids Royale Fortnite Battle Athletic Gift Womens Size Llama Trainers Shoes Mens Sneakers pZxwSEq 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
Flats sandals Leather women Gladiator Sandals Summer Sandals Greek Shoes Leather Gladiator sandals Red Sandals Greek 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 Leather Sandals Sandals Leather Leather Gladiator Greek Shoes sandals women Flats Sandals Summer Gladiator Red Greek Next Page