A running race is held at the end of each summer. After the race, there is a par

Question

A running race is held at the end of each summer. After the race, there is a party held for the runners at which there is a rae. Race participants randomly select an envelope from a large box. The envelopes contain either a thank you note (worth $0) or a gift card (worth $15). There are a total of 20,000 envelopes in the box, 12,000 of the envelopes contain a gift card. A group of 100 teachers from a school district run the race. Suppose that these teachers will be the first to select from the box.

b) Assuming the draws are being made with replacement estimate the chance that the teachers will win over $800. For full credit, you must clearly state (i) the normal density curve for the sum, (ii) standard score calculations and finally (iii) the area calculation.

Explanation / Answer

Let N = total no of envelopes in the box =20000

p= propertion of enevelopes contain gift card = 12000/20000 = 0.6

Define a random variable Y

y = 1 if drawing envelope contain a gift card

= 0 if drawing envelope contain thank you note.

Hence Yi's are i.i.d Bernaulli r.v.s.

n= no. of envelopes are selected =100

X: total no. of envelopes that contain gift card

X = sum(Yi) , i = 1:100

Since draws are being made with replacement.

X follows binomial distribution with parameter n=100 and p=0.6

E(x) = np = 60 = expected number of envelopes that contain gift card.

i.e.Expected winning amount win by teachers =60 * 15 =$900

and V(x) =24

We have to find probabilty that the teachers will over $ 800.

For drawing 1 gift card they win $15. Hence to win $800 to drawn 54 envelopes that contain gift card (800/15=53.33) from the box.

i.e. if the number of envelopes that contain gift card is greater than 53 the expected winning amount greater than $ 800

Required probability = P(X > 53)

Since n=100 ( large) by using normal approximation

X~ B(n,p) then Z=(X-np/sqrt(npq))~N(0,1).

P( x > 53) =P((x-np/sqrt(npq)) > ( 53-60/sqrt(24))

= P(Z > -1.4288)

From normal probability table P(Z >-1.4288) = 0.9235

Hence probability that the teachers will win over $ 800 is 0.9235

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