Question

Johnson Electronics Corporation makes electric tubes. It is known that the standard deviation of the lives of these tubes is 135 hours. The company’s research department takes a sample of 105 such tubes and finds that the mean life of these tubes is 2150 hours. What is the probability that this sample mean is within 20 hours of the mean life of all tubes produced by this company?

Round your answer to four decimal places.

P=___

Answer 1

Here, μ = 2150, σ = 135/sqrt(105) = 13.1747, x1 = 2130 and x2 = 2170. We need to compute P(2130<= X <= 2170). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (2130 - 2150)/13.1747 = -1.52
z2 = (2170 - 2150)/13.1747 = 1.52

Therefore, we get
P(2130 <= X <= 2170) = P((2170 - 2150)/13.1747) <= z <= (2170 - 2150)/13.1747)
= P(-1.52 <= z <= 1.52) = P(z <= 1.52) - P(z <= -1.52)
= 0.9357 - 0.0643
= 0.8714