The Denver Post stated that 70% of all new products introduced in grocery stores

Question

The Denver Post stated that 70% of all new products introduced in grocery stores fail (and are taken off the market) within 2 years. Using normal approximation for this binomial distribution, if a grocery store chain introduces 75 new products

a. Verify that the assumption for normal approximation to the binomial is satisfied.

b. What is the probability that within two years, 54 or more will fail with continuity Correction?

c. What is the probability that within two years, fewer than 62 will fail without continuity Correction?

Explanation / Answer

Please note that IF np > 5 AND nq > 5, then the binomial random variable is approximately normally distributed

here p = 0.7 , q = 1-p = 0.3 and n = 75

so np = 75*0.7 =  52.5

nq = 0.3*75 = 22.5 , hence both the conditions are satisfied . So yes , the assumptions are satified

b) with continutiy corection mean that P(X>54) becomes P(X>54.5) as

If   P(X>n) then we use   P(X > n + 0.5)

so lets use the Z score formula as

Z = (X-Mean)/SD =

now SD = sqrt(npq) = sqrt(75*0.3*0.7) = 3.96

and mean = np = 52.5

Z = (52.5-54.5)/3.96 = -0.505

so from the Z table we get

P ( Z>0.505 )=P ( Z<0.505 )=0.695

c ) now next we need to check P(X<62)

so Z score = (62-54.5)/3.96 = 1.89

So Z score P(Z<1.89)

P ( Z<1.89 )=0.9706 , using the z table

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