4. A personal water craft (PWC) manufacturer wants to compare the potential mark

Question

4. A personal water craft (PWC) manufacturer wants to compare the potential market for its products in the northeastern United States to the market in the southeastern United States. This would help the manufacturer decide where to concentrate their marketing efforts. Using telephone directories, the company randomly chooses 1,000 households in the southeast (SE) and 1,000 households in the northeast (NE) and determines whether each household plans to buy a PWC within the next 5 years. In the southeast, 42h households in the northeast said they would. a) Estimate the difference between the proportions of households planning to purchase a PWC within the next 5 years, using a 95% confidence interval. Conduct a hypothesis test to see if the proportion of households in the SE planning on buying a PWC is more than the proportion of households in the NE. Use the critical value method and D.05. b)

Explanation / Answer

4.

a)

Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.042      
p2 = x2/n2 =    0.024      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.007978722      
          
          
For the   95%   confidence level, then  
          
alpha/2 = (1 - confidence level)/2 =    0.025      
z(alpha/2) =    1.959963985      
Margin of error = z(alpha/2)*sd =    0.015638007      
lower bound = p1^ - p2^ - z(alpha/2) * sd =    0.002361993      
upper bound = p1^ - p2^ + z(alpha/2) * sd =    0.033638007      
          
Thus, the confidence interval is          
          
(   0.002361993   ,   0.033638007 ) [ANSWER]

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b)

Formulating the hypotheses          
Ho: p1 - p2   <=   0  
Ha: p1 - p2   >   0  
Here, we see that pdo =    0   , the hypothesized population proportion difference.  
          
Getting p1^ and p2^,          
          
p1^ = x1/n1 =    0.042      
p2 = x2/n2 =    0.024      
          
Also, the standard error of the difference is          
          
sd = sqrt[ p1 (1 - p1) / n1 + p2 (1 - p2) / n2] =    0.007978722      
          
Thus,          
          
z = [p1 - p2 - pdo]/sd =    2.256000481      
          
As significance level =    0.05   , then the critical z is  
          
zcrit =    1.644853627      
          
As z > 1.645, then we    REJECT THE NULL HYPOTHESIS.  

Hence,   there is significant evidence that the proportion of households in the SE planning on buying a new PWC is more than the proportion of households in the NE at 0.05 level. [CONCLUSION]

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