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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

In Problems 53-56, use a normal approximation.

Revenue Islands Realty estimates that 5% of the families who inquire about time-share villas will purchase one. If this is true, what is the probability that no more than 20 of 500 families who inquire about time-share villas will make a purchase?

To determine

To calculate: The probability of no more than 20 families purchase villas and there are 500 families who ask for the information about the time share villa.

Explanation

Given Information:

There are 500 families who ask for the information about the time share villa.

Formula used:

The compliment of a probability p is,

q=1p

Where, p is a probability of success, q is a probability of not success.

The condition for the approximated normal distribution from binomial probability distribution is,

np5 and nq5

The theoretical mean for the distribution is given as,

μ=np

Where n is the number of independent trials and p is the probability of success.

The standard deviation for the distribution is,

σ=npq

The z- score for the standard deviation σ and theoretical mean μ is,

z=xμσ

Calculation:

The probability of success is the probability of families who inquire for the villa.

p=5100=0.05

Use the formula for the compliment of probability p,

q=1p

Substitute 0.05 for p.

q=10.05=0.95

Apply the normal approximation, the value of theoretical mean np is given as,

np=500(0.05)=25

The value of nq5 is given as,

nq=500(0.95)=475

Since nq5 and nq5 so the binomial distribution can be use as normal approximation distribution.

Use the formula for the theoretical mean,

μ=np

Substitute 500 for n and 0.05 for p,

μ=500(0.05)=25

Use the formula for the standard deviation of a binomial distribution,

σ=npq

Substitute 500 for n, 0

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