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 85% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 66 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

Answer 1

Let X denotes the number of new products introduced in grocery stores fail (are taken off the market) within 2 years.

X ~ Binomial(66, 0.85)

Here n = 66, p = 0.85

Now,

n*p = 66*0.85 = 56.1 > 5

and n*(1 - p) = 66*(1 - 0.85) = 9.9 > 5

So it is appropriate to use the normal approximation to the binomial.

Here

and

Using normal approximation, we can write

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

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

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

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