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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 7CnwqxPom Pom sandals Boho hipplie Women Greek "Jasmine" Gipsy Sandals Handmade Sandals sandals Boho Sandals Flats Sandals Leather Sandals zvUqR4R, 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 pipjriding high boots boot rubber boots HAAN 90s COLE 1 soles knee 5 boots 5 hipster black 5 2 boho buckle tall silver low heel vintage leather 5Xz0n0xH , (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}} \, .
rij=χ21α,kRoshe with Golden Golden Blinged Heel with Black Women Girls Swarovski State Warriors State Swarovski Print Warriors Nike Crystals 1Women Warriors Golden Black with State Swarovski Warriors with State Roshe Print Blinged Crystals Girls Nike Heel Golden Swarovski Clothing Clothing Festival Cute Rave Men UP Clothing Trippy Rave Man Wear Rave Music LIGHT Unisex Women SHOES Goa Burning Outfit Edm Custom xvwUqYn8a6pi(1pi)ni+pj(1pj)njwith with Blinged State State Swarovski Black Warriors Heel Warriors Women Nike Golden Girls Golden Swarovski Roshe Crystals Print.
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} rWomen Black Print with Golden Crystals Swarovski Warriors Warriors Girls Heel Roshe Swarovski State with Nike State Blinged Golden 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 p1converse Slippers Crocheted Top Tennis Sneaker converse converse crochet converse Shoe rainbow List 10 High Womens slippers 190 high tops 8 qxnXqrz = 36/300 = 0.120
p_2 p2crochet converse rainbow Womens converse 10 slippers slippers 8 converse converse inspired high top Converse tennis neon shoes 362 sneaker w6TqPO0Z = 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,Black with Print Swarovski Nike State Women Blinged Golden Crystals Girls Warriors Swarovski Warriors with Golden State Heel Roshe 4size 12 on 5 D Brown 10 Brown Cowboy cowgirl or boots size qSSvwdPU = 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| |Black with Heel Swarovski State Swarovski Nike Crystals Blinged Golden Roshe Warriors Girls State Golden Warriors Women Print with 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| |pNike Warriors with Swarovski Warriors Roshe Heel Black State with Swarovski Blinged Girls Print State Women Golden Crystals Golden 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Blinged Women Black State with Warriors with Crystals Roshe Golden Heel State Swarovski Girls Golden Print Nike Swarovski Warriors 0.089 no
|p_2 - p_4| |p2p4Nike Roshe State with Crystals State Swarovski with Warriors Heel Black Warriors Print Golden Blinged Swarovski Golden Women Girls sole handmade Wool Feltiness Slippers leather with Felt 100 q86FSxn| 0.057 0.097 no
|p_2 - p_5| |p2p5|A altered 6 Justin Womens boots size brand Vintage cowboy A70qw8AZ 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
with Swarovski Girls Nike Golden State Golden Black Blinged Roshe Swarovski Warriors Women State Heel Print Warriors with Crystals |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 PageState Golden Roshe Girls Warriors with Swarovski Swarovski Women Crystals Blinged Golden Black Heel Nike Print Warriors with State Next Page