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.  Is the result from a few years ago likely to be valid today?

Answer 1

$H_0: p_0 = 0.5 \\ H_1: p_0 < 0.5 \\ α=0.01 \\ N= 1500 \\ p = 0.47$

Test-statistic

$Z = \frac{p-p_0}{\sqrt{\frac{p_0(1-p_0)}{N}}} \\ Z = \frac{0.47-0.5}{\sqrt{\frac{0.5(1-0.5)}{1500}}} \\ Z = \frac{-0.03}{\sqrt{\frac{0.5 \times 0.5}{1500}}} \\ Z = -2.324$

At α=0.01 the critical value for a left-tailed test is $Z_c = -2.326$

Since $Z>Z_c$ the null hypothesis is accepted.

There is NOT enough evidence to claim that fewer than half of all households have a high-definition television at 0.01 level of significance.

If the data and claim are the same in few years ago, the results likely to be valid today.