# Nylon Bars Were Tested For Brittleness Each Of 280 Bars Was Moldedunder Similar Cond 4118225

Nylon bars were tested for brittleness. Each of 280 bars was moldedunder similar conditions and was tested in five places. Assumingthat each bar has uniformcomposition, the number of breaks on agiven bar should be binomially distributed with five trials and anunknown probability p of failure. If the bars are allthesame uniform strength, p should vary from bar to bar.Thus, the null hypothesis is that the p's are all equal.The following table summarizes theoutcome of the experiment:

Breaks/BarFrequency0157169
235
3
17
4151
a. Under the given assumption, the data in the table consist of 280observations of independent binomial random variables. Find the mleof p.

b. Pooling the last three cells, test the agreement of the observedfrequency distribution with the binomial distribution usingPearson's chi-square test.

c. Apply the test procedure derived in the previous problem. (LetXi ~ bin(ni, pi)for i = 1, …, m, beindependent. Derive a likelihoodratio test for the hypothesis H0: p1= p2 = … = pm againstthe alternative hypothesis that the piare not all equal. What is the large-sample distribution of thetest statistic?) 4.6/5

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