Question

In an effort to cut costs and improve profits, any US companies have been turning to outsourcing. In fact, according to Purchasing magazine, 54% of companies surveyed outsourced some part of their manufacturing process in the past two to three years. Suppose 555 of these companies are contacted.

1. What is the probability percentage that 338 or more companies outsourced some part of their manufacturing process in the past two or three years? Round the percent to two decimal places. to find answer.
   
2. What is the probability percentage that 285 or more companies outsourced some part of their manufacturing process in the past two or three years? Round the percent to two decimal place. to find answer
   
3. What is the probability percentage that 48% or less of these companies outsourced some part of their manufacturing process in the past two or three years? Round the percent to two decimal places.

Answer 1

Given,

sample n = 555

p = 54% = 0.54

q = 1 - 0.54 = 0.46

Mean = np = 555*0.54 = 299.7

Standard deviation = sqrt(npq) = sqrt(555*0.54*0.46) = 11.74

a)

P(X >= 338) = P((x-u)/s > (338 - 299.7)/11.74)

= P(z > 3.26)

= 0.000557 [since from z table]

= 0.0006

= 0.06%

b)

P(X >= 285) = P((x-u)/s > (285 - 299.7)/11.74)

= P(z > -1.25)

= 0.8943502 [since from z table]

= 0.8944

= 89.44%

c)

P(X < 0.48) = P((x-p)/sqrt(pq/n) < (0.48 - 0.54)/sqrt(0.54(1-0.54)/555))

= P(z < -2.84)

= 0.0022557 [since from z table]

= 0.0023

= 0.23%