USE EXCEL ONLY Use the normal approximation to the binomial distribution to answ

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USE EXCEL ONLY

Use the normal approximation to the binomial distribution to answer this question

In a given hour, 30 individuals enter a certain store. Past experience indicates that 30% of all individuals entering the store decide to make a purchase. Based on the relevant binomial distribution, answer the following questions about X, the number of customers entering the store will make a purchase:

What is the probability X is exactly 3?

What is the probability that X is no more than 10?

What is the probability that X is at least 15?

What is the probability that X is more than 8?

Explanation / Answer

In a given hour, 30 individuals enter a certain store.
Past experience indicates that 30% of all individuals entering the store decide to make a purchase

NORMAL APPROXIMATION TO BINOMIAL DISTRIBUTION
the PDF of normal distribution is = 1/ * 2 * e ^ -(x-u)^2/ 2^2
standard normal distribution is a normal distribution with a,
mean of 0,
standard deviation of 1
mean ( np ) = 30 * 0.3 = 9
standard deviation ( npq )= 30*0.3*0.7 = 2.50998007960223
equation of the normal curve is ( Z )= x - u / sd/sqrt(n) ~ N(0,1)

a.
the probability X is exactly 3
P( X = 3 ) = ( 30 3 ) * ( 0.3^3) * ( 1 - 0.3 )^27
= 0.0072

b.
the probability that X is no more than 10
P(X < 10) = (10-9)/2.51
= 1/2.51= 0.3984
= P ( Z <0.3984) From Standard NOrmal Table
= 0.6548

c.
the probability that X is at least 15
P(X < 15) = (15-9)/2.51
= 6/2.51= 2.3904
= P ( Z <2.3904) From Standard NOrmal Table
= 0.9916
P(X > = 15) = (1 - P(X < 15))
= 1 - 0.99158586056447 = 0.0084

d.
the probability that X is more than 8
P(X > 8) = (8-9)/2.51
= -1/2.51 = -0.3984
= P ( Z >-0.3984) From Standard Normal Table
= 0.6548

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