Question

1.     In a survey of 1500 households, it is found that 47% of them have a high-definition television (based on data from the consumer electronics association). Use a 0.01 significance level to test the claim that fewer than half of all households have a high definition television.  Are the results from a few years ago likely to be valid today?

 

Answer 1

$H_0:p\geqslant 0.5\\H_1:p<0.5\\Z=\sqrt{n}\frac{\hat{p}-p}{\sqrt{p\left( 1-p \right)}}=\sqrt{1500}\frac{0.47-0.5}{\sqrt{0.5\left( 1-0.5 \right)}}=-2.32379\\P-value:\\P\left( Z<-2.32379 \right) =\varPhi \left( -2.32379 \right) =0.0101>0.01\Rightarrow H_0\,\,is\,\,not\,\,rejected\\The\,\,results\,\,from\,\,a\,\,few\,\,years\,\,ago\,\,are\,\,not\,\,likely\,\,to\,\,apply\,\,for\,\,today \,because\,\,the\,\,\\results\,\,in\,\,electronics\,\,area\,\,are\,\,quickly\,\,changing.\\$