Test the following hypotheses for a multinomial probability distribution by using the ${}^2$ goodness of fit test.

$H_0{p}_A=0\mathrm{.}20,p_B=0\mathrm{.}40,$

: and $p_O=0\mathrm{.}40$

$H_2$ The probabilities are not

:$,p_A=0\mathrm{.}20,p_B=0\mathrm{.}40$

and $p_C=0\mathrm{.}40$

A sample of size $200$ yielded $60$ in category A$,120$ in category B$,$ and $20$ in category C$.$ Use $=0\mathrm{.}01$ and test to see whether the probabilities are as stated in $H_0.$

a$.$ Use the $p-$value approach. Use Table $3$ of Appendix B$.$

$x^2=$

$(to2$ decimals$)$

The $p-$value is

Conclusion:

Conclude the proportions differ from $0\mathrm{.}2,0\mathrm{.}4,$ and $0\mathrm{.}4.$

b$.$ Repeat the test using the critical value approach.

$x_{0\mathrm{.}01}^2=(to3$ decimals$)$

$x^2=(to2$ decimals$)$

Conclusion:

Conclude the proportions differ from $0\mathrm{.}2,0\mathrm{.}4,$ and $0\mathrm{.}4.$

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Partially Correct

$0=$ Icon Key