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 70 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

Using Normal Approximation to Binomial
Mean = n * P = ( 70 * 0.8 ) = 56
Variance = n * P * Q = ( 70 * 0.8 * 0.2 ) = 11.2
Standard deviation = √(variance) = √(11.2) = 3.3466

Condition check for Normal Approximation to Binomial
n * P >= 10 = 70 * 0.8 = 56
n * (1 - P ) >= 10 = 70 * ( 1 - 0.8 ) = 14

Part a)

P ( X >= 47 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 47 - 0.5 ) =P ( X > 46.5 )

X ~ N ( µ = 56 , σ = 3.3466 )
P ( X > 46.5 ) = 1 - P ( X < 46.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 46.5 - 56 ) / 3.3466
Z = -2.84
P ( ( X - µ ) / σ ) > ( 46.5 - 56 ) / 3.3466 )
P ( Z > -2.84 )
P ( X > 46.5 ) = 1 - P ( Z < -2.84 )
P ( X > 46.5 ) = 1 - 0.0023
P ( X > 46.5 ) = 0.9977

Part b)

P ( X <= 58 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 58 + 0.5 ) = P ( X < 58.5 )

X ~ N ( µ = 56 , σ = 3.3466 )
P ( X < 58.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 58.5 - 56 ) / 3.3466
Z = 0.75
P ( ( X - µ ) / σ ) < ( 58.5 - 56 ) / 3.3466 )
P ( X < 58.5 ) = P ( Z < 0.75 )
P ( X < 58.5 ) = 0.7734

Part c)

Using Normal Approximation to Binomial
Mean = n * P = ( 70 * 0.2 ) = 14
Variance = n * P * Q = ( 70 * 0.2 * 0.8 ) = 11.2
Standard deviation = √(variance) = √(11.2) = 3.3466

Condition check for Normal Approximation to Binomial
n * P >= 10 = 70 * 0.2 = 14
n * (1 - P ) >= 10 = 70 * ( 1 - 0.2 ) = 56

P ( X >= 15 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 15 - 0.5 ) =P ( X > 14.5 )

X ~ N ( µ = 14 , σ = 3.3466 )
P ( X > 14.5 ) = 1 - P ( X < 14.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 14.5 - 14 ) / 3.3466
Z = 0.15
P ( ( X - µ ) / σ ) > ( 14.5 - 14 ) / 3.3466 )
P ( Z > 0.15 )
P ( X > 14.5 ) = 1 - P ( Z < 0.15 )
P ( X > 14.5 ) = 1 - 0.5596
P ( X > 14.5 ) = 0.4404

Part d)

P ( X < 10 )
Using continuity correction
P ( X < n - 0.5 ) = P ( X < 10 - 0.5 ) = P ( X < 9.5 )

X ~ N ( µ = 14 , σ = 3.3466 )
P ( X < 9.5 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 9.5 - 14 ) / 3.3466
Z = -1.34
P ( ( X - µ ) / σ ) < ( 9.5 - 14 ) / 3.3466 )
P ( X < 9.5 ) = P ( Z < -1.34 )
P ( X < 9.5 ) = 0.0901