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 80% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 61 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

Normal Approximation to Binomial Distribution
Mean ( np ) =61 * 0.8 = 48.8
Standard Deviation ( npq )= 61*0.8*0.2 = 3.1241
Normal Distribution = Z= X- u / sd                   
a.
P(X >= 47) = (47-48.8)/3.1241
= -1.8/3.1241 = -0.5762
= P ( Z >-0.576) From Standard Normal Table
= 0.7177                  
b.
P(X > 58) = (58-48.8)/3.1241
= 9.2/3.1241 = 2.9448
= P ( Z >2.945) From Standard Normal Table
= 0.0016                  
c.
P(X >= 15) = (15-12.2)/3.1241
= 2.8/3.1241 = 0.8963
= P ( Z >0.896) From Standard Normal Table
= 0.1851  
d.
P(X < 10) = (10-12.2)/3.1241
= -2.2/3.1241= -0.7042
= P ( Z <-0.7042) From Standard NOrmal Table
= 0.2407                  
              

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.