1. The Denver Post reported that 80% of all new products introduced in grocery s

Question

1. The Denver Post reported that 80% of all new products introduced in grocery stores fail within 2 years. If a grocery store chain introduces 81 new products, what is the probability that within 2 years exactly 60 will fail?

2. You are trying to determine if you should accept a shipment of eggs for a local grocery store. About 4% of all cartons which are shipped have had an egg crack while traveling. You are instructed to accept the shipment if no more than 10 cartons out of the 300 you inspect have cracked eggs. What is the probability that you accept the shipment? (In other words, what is the probability that, at the most, you had 10 cartons with cracked eggs?)

3. Market researcher Gartner Dataquest reports that Dell Computer controls 25% of the PC market in the United States. Suppose a business researcher randomly selects 192 recent purchasers of PCs. What is the probability that more than 58 purchasers bought a Dell computer?

4. In which of the following situations can you approximate the binomial probability distribution with the given values of p and n by using a normal distribution?

p = .16 and n = 30

p = .20 and n = 36

p = .65 and n = 10

p = .90 and n = 40

More than one of the above

p = .16 and n = 30

p = .20 and n = 36

p = .65 and n = 10

p = .90 and n = 40

More than one of the above

Explanation / Answer

Q1.
Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X = 60 ) = ( 81 60 ) * ( 0.8^60) * ( 1 - 0.8 )^21
= 0.0438

Q2.

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

P( X < = 10) = P(X=10) + P(X=9) + P(X=8) + P(X=7) + P(X=6) + P(X=5) + P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)
= ( 300 10 ) * 0.04^10 * ( 1- 0.04 ) ^290 + ( 300 9 ) * 0.04^9 * ( 1- 0.04 ) ^291 + ( 300 8 ) * 0.04^8 * ( 1- 0.04 ) ^292 + ( 300 7 ) * 0.04^7 * ( 1- 0.04 ) ^293 + ( 300 6 ) * 0.04^6 * ( 1- 0.04 ) ^294 + ( 300 5 ) * 0.04^5 * ( 1- 0.04 ) ^295 + ( 300 4 ) * 0.04^4 * ( 1- 0.04 ) ^296 + ( 300 3 ) * 0.04^3 * ( 1- 0.04 ) ^297 + ( 300 2 ) * 0.04^2 * ( 1- 0.04 ) ^298 + ( 300 1 ) * 0.04^1 * ( 1- 0.04 ) ^299 + ( 300 0 ) * 0.04^0 * ( 1- 0.04 ) ^300
= 0.3429

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