Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities. Do you take the free samples offered in supermarkets? About 58% of all customers will take free samples. Furthermore, of those who take the free samples, about 33% will buy what they have sampled. Suppose you set up a counter in a supermarket offering free samples of a new product. The day you were offering free samples, 315 customers passed by your counter. (Round your answers to four decimal places.) (a) What is the probability that more than 180 will take your free sample? (b) What is the probability that fewer than 200 will take your free sample? (c) What is the probability that a customer will take a free sample and buy the product? Hint: Use the multiplication rule for dependent events. Notice that we are given the conditional probability P(buy|sample) = 0.33, while P(sample) = 0.58. (d) What is the probability that between 60 and 80 customers will take the free sample and buy the product? Hint: Use the probability of success calculated in part (c).

Answer 1

np =    315*0.58=       182.70 ≥10

n(1-p) = 132.30 ≥10

so, normal approx can be assumed

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a)

Sample size , n =    315          
Probability of an event of interest, p =   0.58          

Mean = np =    182.7          
std dev ,σ=√np(1-p)=   8.7598          
              
P(X >   180   )      
              
Z=(X - µ ) / σ =        (180-182.7)/8.7598)=       -0.308
              
=P(Z >   -0.308   ) = 0.6210  

b)
              
P(X ≤   200   )      
              
Z=(X - µ ) / σ =        (200-182.7)/8.7598)=       1.975
              
=P(Z≤   1.975   ) =    0.9759  

c) p=0.33*0.58=0.1914

d)

Sample size , n =    315              
Probability of an event of interest, p =   0.1914              

Mean = np =    60.291              
std dev ,σ=√np(1-p)=   6.9822              
                  
P (   60   ≤ X ≤    80   )  
Z1 =   (X1 - µ ) / σ =   -0.042          
Z2 =   (X2- µ ) / σ =   2.823          
                  
P (    -0.0417   ≤ Z ≤   2.823   )   
= P ( Z ≤   2.823   ) - P ( Z ≤   -0.042   ) =   
=   0.9976   -    0.4834   =   0.5142

please revert for doubts..