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 78% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 63 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 within 2 years out of 63 selected new products.

X ~ Binomial(63, 0.78)

E(X) = 63*0.78 = 49.14

and

Here

n*p = 63*0.78 = 49.14 >=5

n*q = 63*(1-0.78) = 13.86 >=5,

So, in this case it is appropriate to use the normal approximation to the binomial.

Using normal approximation,

X ~ Normal(49.14, 3.28798²)

Now

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