Question

Consider a multinomial experiment with n=400n=400 and k=3k=3. The null hypothesis is H0:p1=0.60,p2=0.25H_0: p_1=0.60, p_2=0.25, and p3=0.15p_3=0.15. The observed frequencies resulting from the experiment are

 Category 123 Frequency 2509456\begin{array}{|l|r|r|r|} \hline \text { Category } & 1 & 2 & 3 \\ \hline \text { Frequency } & 250 & 94 & 56 \\ \hline \end{array}

a. Define the alternative hypothesis.

b. Calculate the value of the test statistic and approximate the pp-value for the test.

c. At the 5% significance level, what is the conclusion to the hypothesis test?

Solution

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a.)\bold{a.)} The null hypothesis states that

HA:All population proportions are equal to their predicted values.H_A:\text{All population proportions are equal to their predicted values.}

so the alternative hypothesis will be the opposite.

HA:Not all population proportions are equal to their predicted values.H_A:\text{Not all population proportions are equal to their predicted values.}

The predicted values is equal to the proportion multiplied to nn. So in Mathematical terms, the null hypotheses is

H0:p1=0.60,p2=0.25,p3=0.15H_0: p_1=0.60, p_2=0.25, p_3=0.15

and the Alternative hypotheses is

Ha:p10.60 or p20.25 or p30.15H_a: p_1\ne0.60 \text{ or } p_2\ne0.25 \text{ or } p_3\ne0.15

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