A humane society claims that less than 37% of households own a dog. in a random

Question

A humane society claims that less than 37% of households own a dog. in a random sample of 402 households, 155 say they own a dog. At a=0.04 ,is there enough evidence tho support society's claim? a) write the claim mathematically and identify Ho and Ha. b) find critical value, c) find the standardized test stastic d) decide whether to reject or fail to reject the null hypothesis, e) Interpret the decision made in the context of the original claim.    I already figured out A, just need rest, thankyou.

Explanation / Answer

let p be the proportion of households own a dog

claim is that p=37%

we have a sample of n=402 households out of which t=155 own a dog

level of significance=alpha=0.04

A humane society claims that less than 37% of households own a dog

a) so H0:p=0.37 Ha: p<0.37

b) since n=402 is very high one sample z test is applicable here

and since the alternative hypothesis is left sided

hence the critical value is -tao0.04=-tao0.04 , the upper 0.04 point of a standard normal distribution.

=-1.750686 [answer]

c) let phat be the sample proportions of households that own a dog

then E(phat)=p V[phat]=p(1-p)/n

but under H0 p=0.37 n=402

hence the standardized test statistic is T=(phat-0.37)/sqrt[0.37*(1-0.37)/n] which under H0 follows N(0,1) approximately.

Ho is rejected iff t<-tao0.02

d) now phat=155/402=0.38557

hence the observed value of T is t=(0.38557-0.37)/sqrt[0.37*(1-0.37)/402]=0.64659

critical value is -1.750686

hence t>-1.750686

hence we fail to reject the null hypothesis.

e) hence at 4% level of significance it is interpreted that the claim of the humane society that less than 37% of households own a dog is false

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