Question

Only 4% of items produced by a machine are defective. A random sample of 200 items is selected and checked for defects.

a. Refer to Exhibit 7-1. What is the expected value for ?

b. What is the probability that the sample proportion will be within +/-0.03 of

the population proportion

c.What is the probability that the sample proportion will be between 0.04 and

0.07?

Answer 1

a)

Expected value for proportion = 0.04

b)

Here, μ = 0.04, σ = 0.0139, x1 = 0.01 and x2 = 0.07. We need to compute P(0.01<= X <= 0.07). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (0.01 - 0.04)/0.0139 = -2.16
z2 = (0.07 - 0.04)/0.0139 = 2.16

Therefore, we get
P(0.01 <= X <= 0.07) = P((0.07 - 0.04)/0.0139) <= z <= (0.07 - 0.04)/0.0139)
= P(-2.16 <= z <= 2.16) = P(z <= 2.16) - P(z <= -2.16)
= 0.9846 - 0.0154
= 0.9692

c)

Here, μ = 0.04, σ = 0.0139, x1 = 0.04 and x2 = 0.07. We need to compute P(0.04<= X <= 0.07). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (0.04 - 0.04)/0.0139 = 0
z2 = (0.07 - 0.04)/0.0139 = 2.16

Therefore, we get
P(0.04 <= X <= 0.07) = P((0.07 - 0.04)/0.0139) <= z <= (0.07 - 0.04)/0.0139)
= P(0 <= z <= 2.16) = P(z <= 2.16) - P(z <= 0)
= 0.9846 - 0.5
= 0.4846