Of all customers purchasing automatic garage door openers, only one out of every
ID: 2922557 • Letter: O
Question
Of all customers purchasing automatic garage door openers, only one out of every 40 purchase an imported Swedish model. It is very costly for the company to stock these and they want to determine whether to keep the Swedish model on their shelves. A sample of n = 500 customers purchasing automatic garage door openers are asked about their purchase. Let X = number of customers in the sample who purchase the Swedish model. (a) Identify the distribution of X along with its parameters. (b) Identify a distribution that can be used to approximate the true distribution of X. Verify conditions for the approximation to be valid. (c) Regardless of your conclusion in (b), use the approximation to cal- fo culate the probability that more than 12 customers by the Swedish model.Explanation / Answer
a) by defining the random variable
X= Number of customers who purchase the swedish model
p= propertion( Probabilty of sucess) of customers who purchase the swedish model = 1/40 = 0.025
n= sample size = 500
The random variable X follows binomial distribution with parameter n =500 and p= 0.025
X ~ B( n=500 p=0.025) (Answer)
b) Since p= 0.025 ( Very small) and n=500 (very large) then binomial distribution tends to normal distribution with mean np =12.5 and variance = npq = 12.1875
Bu using Normal approximation X ~ N ( 12.5, 12.1875)
c) To find the probability that the more than 12 cutomers purchace the swedish model
required probability =P( X>12)
P(X>12) = P ( (X- mean/standard deviation)> (12- 12.5/3.4910))
= P (Z >-0.1432) (Z follows standard normal distribution with mean 0 and variance 1)
by using normal probabilty
P(Z> -0.1432)=0.5557
Answer : Probability that more than 12 customers by the swedish model is 0.5557
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