In the following problem, check that it is appropriate to use the normal approxi

Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

It is known that 76% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 62 new products, find the following probabilities. (Round your answers to four decimal places.)

(a) within 2 years 47 or more fail


(b) within 2 years 58 or fewer fail


(c) within 2 years 15 or more succeed


(d) within 2 years fewer than 10 succeed

Explanation / Answer

Solution:-

p = 0.76

a) The probability that within 2 years 47 or more fail is 0.58284.

x = 47

By applying binomial distribution:-

P(x,n) = nCx*px*(1-p)(n-x)

P(x > 47) = 0.58284.

b) The probability that within 2 years 58 or fewer fail is 0.000283.

x = 58

By applying binomial distribution:-

P(x,n) = nCx*px*(1-p)(n-x)

P(x > 58) = 0.000283.

c) The probability that within 2 years 15 or more succeed is 0.5347.

p = 1 - 0.76 = 0.24

By applying binomial distribution:-

P(x,n) = nCx*px*(1-p)(n-x)

P(x > 15) = 0.5347

d) The probability that within 2 years fewer than 10 succeed is 0.04935.

p = 0.24

By applying binomial distribution:-

P(x,n) = nCx*px*(1-p)(n-x)

P(x < 10) = 0.04935

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