32 HDTV Penetration In a survey of 1500 households it is fou

32.

HDTV Penetration 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?

Solution

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   >=   0.5
Ha:   p   <   0.5
As we see, the hypothesized po =   0.5      
As it is given,          
          
po =    0.5      
          
Getting the standard error of po, spo,          
          
spo = sqrt[po (1 - po) / n] =    0.012909944      
          
Getting the z statistic,          
p^ =    0.47      
z = (p^ - po)/spo =    -2.323790008      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.010068376      

significance level =    0.01      

Comparing p and the significance value, we   FAIL TO REJECT THE NULL HYPOTHESIS.      

Thus, there is no significant evidence that fewer than half of all households have a high-definition television.

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The result from few years ago will NOT be likely to be valid today, as HD TVs are more common nowadays.