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## 7.4.7.4. |
## Comparing multiple proportions: The Marascuillo procedure |

* Testing for equal proportions of defects *
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* 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
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, (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}} \, .
*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}
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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
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= 36/300 = 0.120

p_2
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= 46/300 = 0.153

p_3
p3
= 42/300 = 0.140

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= 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
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= 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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