In a survey of 1500 households it was learned that 47 of the

In a survey of 1500 households, it was learned that 47% of them have a high definition TV. Using a 0.01 significance level, test the claim that fewer than half of all households have a high definition TV.

Solution

The test hypothesis:

Ho: p=0.5 (i.e. null hypothesis)

Ha: p<0.5 (i.e. alternative hypothesis)

The test statistic is

Z=(phat-p)/sqrt(p*(1-p)/n)

=(0.47-0.5)/sqrt(0.5*0.5/1500)

=-2.32

It is a left-tailed test.

Given a=0.01, the critical value is Z(0.01) =-2.33 (from standard normal table)

The rejection region if Z<-2.33, we reject the null hypothesis.

Since Z=-2.32 is larger than -2.33, we do not reject the null hypothesis.

So we can not conclude that fewer than half of all households have a high definition TV