## Related questions with answers

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

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

a. Define the alternative hypothesis.

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

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

Solution

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

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

so the alternative hypothesis will be the opposite.

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

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

$H_0: p_1=0.60, p_2=0.25, p_3=0.15$

and the Alternative hypotheses is

$H_a: p_1\ne0.60 \text{ or } p_2\ne0.25 \text{ or } p_3\ne0.15$

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