Verify that the formulas for the \(\chi^{2}\) statistic on page 311 (with \(\widehat{p}\) substituted for the \(p_{i}\) ) and on page 312 are equivalent.

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EXAMPLE 8 Testing the equality of three proportions using the x statistic Samples of three kinds of materials, subjected to extreme temperature changes, pro- duced the results shown in the following table: Material A Material B Material C Total Crumbled 41 27 22 90 Remained intact 79 53 78 210 Total 120 80 100 300 Use the 0.05 level of significance to test whether, under the stated conditions, the probability of crumbling is the same for the three kinds of materials.