Question

Sampling Distribution of Proportions A restaurant chain regularly surveys its customers. On the basis of these surveys, the management of the chain claims that 75% of its customers rate the food as excellent. A consumer testing service wants to examine this claim. They take a simple random sample of 600 customers and ask them each to rate the food. Assume that the restaurant chain’s claim is true (i.e. that the true proportion is indeed 0.75).

a) assuming that the chain’s claim is true (i.e. that the true proportion is indeed 0.75), what is the probability that, in a simple random sample of 600 customers, less than 70% rate the food as excellent?

b)Suppose that in a random sample of 600 customers, 420 rate the food as excellent. Does this support or refute the restaurant chain’s claim? Hint: Refer to the previous question.

c) if the process is repeated with 1000 customers, what is the probability that 70% rate the food excellent?

Answer 1

A)

population proportion ,p=   0.75                      
n=   600                      
                          
std error , SE = √( p(1-p)/n ) =    0.0177                      
                          
sample proportion , p̂ =   0.70                      
Z=( p̂ - p )/SE= (   0.700   -   0.75   ) /    0.018   =   -2.828
P ( p̂ <    0.700   ) =P(Z<( p̂ - p )/SE) =                  
                          
=P(Z <    -2.828   ) =    0.0023              

B)

Ho :   p =    0.75                  
H1 :   p ╪   0.75       (Two tail test)          
                          
Level of Significance,   α =    0.05                  
Number of Items of Interest,   x =   420                  
Sample Size,   n =    600                  
                          
Sample Proportion ,    p̂ = x/n =    0.7000                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0177                  
Z Test Statistic = ( p̂-p)/SE = (   0.7000   -   0.75   ) /   0.0177   =   -2.8284
                                
p-Value   =   0.00467 [excel formula =2*NORMSDIST(z)]              
Decision:   p-value<α , reject null hypothesis

CHAIN'S CLAIM IS NOT ACCEPTED

C)

population proportion ,p=   0.75                      

Sample Size,   n =    1000                  
                          
Sample Proportion ,    p̂ = 0.70   
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0137                  
Z Test Statistic = ( p̂-p)/SE = (   0.7000   -   0.75   ) /   0.0137   =   -3.6515
                          
  
p(Z = -3.6515) =   0.00026
  

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