A marketing analyst for a manufacturer of household products is trying to predic

Question

A marketing analyst for a manufacturer of household products is trying to predict the market share for the company’s new brand of floor cleaner that “cleans and waxes in one easy step”. Based on past experience with other new products that the company has developed, the vice president (VP) of product development feels that the population proportion of consumers who will purchase the new cleaner is p=.20 (that is, a 20% market share). The analyst plans to select a random sample of 400 consumers, and determine the sample proportion of consumers who indicate that they will purchase this product when it becomes available.

A) If the sample proportion determined by the analyst is greater than .245, she will consider this to be evidence that the population proportion of consumers who will purchase this product is at least p=.20, and as such, she will recommend to the vice-president that no advertising needs to accompany the release of the product. If it is TRUE that the population proportion of consumers who will buy this product is p=.20, what is the probability that the sample proportion determined by the analyst will be greater than .245, resulting in no expenditure on advertising?

B) The VP would like the analyst to use a larger sample size than 400 consumers to determine the sample proportion described above. The VP suggests using a sample of 800 consumers to compute the sample proportion of consumers who indicate they will purchase the new cleaner. If it is TRUE that p = .20, what effect will the increase in sample size have on the probability that the analyst will observe a sample proportion greater than .245, resulting in no expenditure on advertising? State your conclusion clearly.

Explanation / Answer

a)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    0.245      
u = mean = p =    0.2      
          
s = standard deviation = sqrt(p(1-p)/n) =    0.02      
          
Thus,          
          
z = (x - u) / s =    2.25      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   2.25   ) =    0.012224473 [ANSWER]

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b)

We first get the z score for the critical value. As z = (x - u) / s, then as          
          
x = critical value =    0.245      
u = mean = p =    0.2      
          
s = standard deviation = sqrt(p(1-p)/n) =    0.014142136      
          
Thus,          
          
z = (x - u) / s =    3.181980515      
          
Thus, using a table/technology, the right tailed area of this is          
          
P(z >   3.181980515   ) =    0.000731358

Hence, as we can see, the probability REDUCED, because of the reduction in variation in the sample proportion. [ANSWER: The probability reduced/decreased]

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