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 90 GPa and 2.1 GPa, respectively. Suppose the distribution is normal.

a) Calculate P(89 ≤ X ≤ 91) when n = 9.

b) How likely is it that the sample mean diameter exceeds 91 when n = 25?

Answer 1

a)

X ~ N ( µ = 90 , σ = 2.1 )
P ( 89 < X < 91 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 89 - 90 ) / ( 2.1 / √(9))
Z = -1.43
Z = ( 91 - 90 ) / ( 2.1 / √(9))
Z = 1.43
P ( 89 < X̅ < 91 ) = P ( Z < 1.43 ) - P ( Z < -1.43 )
P ( 89 < X̅ < 91 ) = 0.9236 - 0.0764 (from Z table)
P ( 89 < X̅ < 91 ) = 0.8473

b)

X ~ N ( µ = 90 , σ = 2.1 )
P ( X > 91 ) = 1 - P ( X < 91 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 91 - 90 ) / ( 2.1 / √ ( 25 ) )
Z = 2.38
P ( ( X - µ ) / ( σ / √ (n)) > ( 91 - 90 ) / ( 2.1 / √(25) )
P ( Z > 2.38 )
P ( X̅ > 91 ) = 1 - P ( Z < 2.38 )
P ( X̅ > 91 ) = 1 - 0.9913 (From Z table)
P ( X̅ > 91 ) = 0.0087