According to a survey, 22% of households relied solely on their cell phones for

Question

According to a survey, 22%
of households relied solely on their cell phones for phone service instead of landlines. It was also reported that this percentage has steadily increased over the previous years. A phone company would like to sample 161
households randomly to test the hypothesis that the proportion of cell phone-only households increased. Complete parts a through d below. Using =0.01,
calculate the probability of a Type II error occurring if the actual proportion of cell phone-only households is 0.31.

Explanation / Answer

here null hypothesis:Ho:p<=0.22

alternate hypothesis:Ha:p>0.22

for p=0.22 and n=161 ; std error of mean =(p(1-p)/n)1/2 =0.0326

for 0.01 level and right tailed test ; critical z =2.326

hence acceptance region =P(X<0.22+2.326*0.0326)=P(X<0.2959)

for true proportion p=0.31 ; std error =(p(1-p)/n)1/2 =0.0364

hence type II error =P(X<0.2959)=P(Z<(0.2959-0.31)/0.0364)=P(Z<-0.3855) =0.3499

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