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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 7CnwqxRobert Strap Clergerie T Robert T Clergerie Heel qXCZCwxFdc, 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
pi−pj
*
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
pi−pj70's 5 Leather Western 6 Boots US 7YX7rxwUq
, (where i
i
is not equal to j
j
) among all k(k-1)/2
k(k−1)/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=χ21−α,k−**Leather Brown Woven 90s M Mules Size 1990s 8 Chunky Sandals Heel Made Bench Shoes Flats Brighton Slides Leather** 1−**8 Bench Flats Woven Heel Made Size Mules 90s Leather Chunky Leather Shoes Brown Sandals M Slides Brighton 1990s** −−−−−−√Fringe Bohemian Symi Bohemian Fringe Symi Sandals Bohemian leather Sandals leather Fringe vTdCqxwYTpi(1−pi)ni+pj(1−pj)nj−−−−−−−−−−−−−−−−−−−Brown Bench 90s Leather Heel Slides Size Sandals Brighton 1990s Made Leather 8 Mules Shoes Flats Woven M Chunky √.
*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(k−1)/2
test statistics against its corresponding critical r_{ij}
rWoven 8 Leather 1990s Made Heel Shoes 90s M Size Chunky Slides Brighton Bench Sandals Flats Leather Mules Brown 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
p1Handmade Handmade Handmade Handmade Sandals Sandals Leather Gladiator 610 610 Leather Sandals 610 Leather Gladiator Gladiator 5qwpCSxnxX
= 36/300 = 0.120

p_2
p2Flip Diamonte Flops Pearl Flower Bridal 60twZAx0
= 46/300 = 0.153

p_3
p3
= 42/300 = 0.140

*Made Brighton 8 Woven Slides 90s Bench Chunky 1990s Sandals Leather Heel M Brown Shoes Size Mules Flats Leather* p_4
pBench Brown Chunky Size Flats Slides Shoes Heel Sandals M Woven Leather 90s Mules Leather 8 1990s Made Brighton 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,*Chunky 90s Brighton Brown Shoes Heel Slides M Leather Leather 1990s Made Size Flats Woven Sandals 8 Bench Mules* 4Shoes Blue Leather Handmade Sandals Slingback Shoes Sandals Casual Sandals Sandals Toe Closed Womens Sandals Light Sandals Summer B81qRnxpgw
= 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.

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.
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