Question

Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for metal sheets of a particular type, its mean value and standard deviation are 75 GPa and 2.4 GPa,respectively. Suppose the distribution is normal. (Round your answers to four decimal places.)

a. calculate P(74 ≤ X ≤ 76) when n = 25.

b. How likely is it that the sample mean diameter exceeds 76 when n = 36?

Answer 1

a)

Here, μ = 75, σ = 0.48, x1 = 74 and x2 = 76. We need to compute P(74<= X <= 76). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (74 - 75)/0.48 = -2.08
z2 = (76 - 75)/0.48 = 2.08

Therefore, we get
P(74 <= X <= 76) = P((76 - 75)/0.48) <= z <= (76 - 75)/0.48)
= P(-2.08 <= z <= 2.08) = P(z <= 2.08) - P(z <= -2.08)
= 0.9812 - 0.0188
= 0.9624

b)

Here, μ = 75, σ = 0.4 and x = 76. We need to compute P(X >= 76). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (76 - 75)/0.4 = 2.5

Therefore,
P(X >= 76) = P(z <= (76 - 75)/0.4)
= P(z >= 2.5)
= 1 - 0.9938 = 0.0062