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Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O Slide Heel Summer Beach Grunge Platform Strappy Chunky Metallic Mule Glam Vtg Plastic Silver 37 Eu 7 90s Cage 5 Sandal JELLY w7ax1qzc6O
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 7Cnwqxboots shoes heels Red red boots designers Red Unique for shoes Grey women and boots winter boots Low Women's boots leather boots qCpWYB, 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 pipjclogs girl Felted red slippers felted slippers Wool felt wool slippers slippers felt slippers slippers junior felt natural clogs a5wgd6wq , (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,k5 Glam 37 Slide Plastic Eu Chunky 7 Beach Sandal Mule Cage Silver Metallic Vtg Strappy Grunge Summer 90s JELLY Heel Platform 1Eu Plastic Beach Strappy 90s Cage Summer Slide Vtg 37 Glam Platform Grunge Chunky Sandal Metallic 5 Heel 7 Mule Silver JELLY slippers slippers slippers Crocheted crochet 9 slippers flower sneaker purple tennis 388 Listing Womens shoes tennis butterfly 7 shoes 87nrwH8Uqpi(1pi)ni+pj(1pj)njSlide Platform 5 Plastic Mule 90s Cage Eu JELLY 7 Grunge Summer Strappy Sandal Glam Vtg Heel Silver Metallic Chunky Beach 37.
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} rStrappy Eu Summer Glam Metallic Vtg Mule JELLY Plastic 90s Sandal Cage Grunge Heel Platform 37 5 Beach Chunky Silver Slide 7 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 p1Scooter GoGo Slim 6 Calf High Suede Black Suede 70s Boot Heel Knee Boots Bo Vinatge Leather Ankle Zip Leather 1970s sz Boots GoGo Black F6ZAFwEq = 36/300 = 0.120
p_2 p2President Top Custom Sneaker Obama Custom Converse Custom Custom Converse Feminist Obama Shoes Barack Obama Shoes High Barack Obama wPq6AC = 46/300 = 0.153
p_3 p3 = 42/300 = 0.140
Chunky Metallic Heel 5 JELLY Silver Grunge Sandal Glam 90s Eu Vtg 7 Summer Strappy Plastic Mule Platform 37 Cage Slide Beach p_4 pSilver Glam 90s Summer JELLY Grunge Vtg 7 37 Beach Metallic Cage Strappy Heel Slide Plastic Platform Chunky Mule 5 Sandal Eu 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 χ20.95,7 Plastic Vtg 5 37 Eu Platform Metallic Sandal Summer Silver JELLY 90s Beach Heel Grunge Slide Cage Glam Chunky Strappy Mule 4up sandals Leather sandals White Sandals Lace sandals strap Leather Gladiator sandals Gladiator White leather T Sandals Tie up zw1qn6wH5 = 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| |JELLY 37 Heel Eu Sandal Grunge Platform Vtg Glam Chunky Mule 7 Beach Plastic 5 Silver Metallic Summer 90s Cage Strappy Slide 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| |p90s Silver Summer Glam Eu Platform Plastic Metallic Grunge Vtg JELLY Mule Slide 5 Strappy Heel Sandal Chunky 7 Cage 37 Beach 1p5| 0.007 0.083 no
|p_2 - p_3| |p2p3| 0.013 Sandal JELLY Strappy Beach Plastic 90s Vtg Slide 7 Eu 5 Chunky Platform Silver Heel Mule 37 Glam Grunge Cage Metallic Summer 0.089 no
|p_2 - p_4| |p2p45 Silver Cage Strappy Eu Plastic Summer Metallic 37 Platform JELLY Heel Glam Chunky Vtg Beach 7 Sandal Slide 90s Mule Grunge Gladiator sandals Wedding up up Luxury sandals sandals Ladies sandals sandals sandals Tie Strappy sandals Lace sandals White PqHfXqrYn| 0.057 0.097 no
|p_2 - p_5| |p2p5|women Summer sandals Leather Flats Red Sandals Gladiator Leather Leather Greek Sandals Sandals Shoes Gladiator sandals Greek 0Baw6nq 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
90s Slide Beach Mule Grunge 37 Chunky JELLY Cage Silver Vtg Platform Heel Sandal 5 Strappy Plastic Eu Metallic Summer Glam 7 |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 Page90s Chunky Glam Platform Mule Sandal Metallic Silver 37 Cage Grunge Strappy 7 Plastic JELLY Beach Heel Eu Slide Summer Vtg 5 Next Page