Of all customers purchasing automatic garage door openers, only one out of every

Question

Of all customers purchasing automatic garage door openers, only one out of every 140 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 = 420 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 calculate the probability that more than 2 customers by the Swedish model.

Explanation / Answer

Normal approximation to binomial distribution

Of all customers purchasing automatic garage door openers, only one out of every 140 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 = 420 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.

Solution:

Here, the distribution of X is given as binomial distribution with parameters n = 420 and p = 1/140.

(b) Identify a distribution that can be used to approximate the true distribution of X. Verify conditions for the approximation to be valid.

Solution:

Here, we have to use a normal distribution approximation to a true binomial distribution. We have to check two conditions whether np>5 and nq>5

We are given, n = 420 and p = 1/140

First condition = np = 420*(1/140) = 420*0.007143 = 3 < 5

First condition fails.

Second condition = nq = 420*(1 – (1/140)) = 420*(1 – 0.007143) = 420*0.992857 = 417

Second condition proves.

One condition is fails to use the normal approximation to the binomial distribution.

(c) Regardless of your conclusion in (b), use the approximation to calculate the probability that more than 2 customers by the Swedish model.

Solution:

Here, we have

Mean = np = 420*(1/140) = 420*0.007143 = 3

Standard deviation = sqrt (npq) = sqrt (420*0.007143*0.992857) = 1.725854

We have to find P(X>2)

P(X>2) = 1 – P(X<2)

Z = ( X – mean ) / standard deviation

Z = (2 – 3) / 1.725854 = -0.57942

P(X<2) = P(Z<-0.57942) = 0.281152

P(X>2) = 1 – P(X<2) = 1 – P(Z<-0.57942) = 1 – 0.281152 = 0.718848

Required probability = 0.718848

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