Question

Suppose x has a distribution with a mean of 70 and a standard deviation of 4. Random samples of size n = 64 are drawn.

(a) Describe the x distribution and compute the mean and standard deviation of the distribution. x has distribution with mean μx = and standard deviation σx = .

(b) Find the z value corresponding to x = 71. z =

(c) Find P(x < 71). (Round your answer to four decimal places.) P(x < 71) =

(d) Would it be unusual for a random sample of size 64 from the x distribution to have a sample mean less than 71?

Explain. No, it would not be unusual because less than 5% of all such samples have means less than 71. Yes, it would be unusual because less than 5% of all such samples have means less than 71. Yes, it would be unusual because more than 5% of all such samples have means less than 71. No, it would not be unusual because more than 5% of all such samples have means less than 71.

Answer 1

Part a)

µ_X̅ = µ = 70
σ_X̅ = σ / √ (n) = 4/√64 = 0.5

Part b)

X ~ N ( µ = 70 , σ = 4 )

Standardizing the value
Z = ( X - µ ) / σ
Z = ( 71 - 70 ) / 4
Z = 0.25

Part c)

X ~ N ( µ = 70 , σ = 4 )
P ( X < 71 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 71 - 70 ) / 4
Z = 0.25
P ( ( X - µ ) / σ ) < ( 71 - 70 ) / 4 )
P ( X < 71 ) = P ( Z < 0.25 )
P ( X < 71 ) = 0.5987

Part d)

X ~ N ( µ = 70 , σ = 4 )
P ( X < 71 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 71 - 70 ) / ( 4 / √64 )
Z = 2
P ( ( X - µ ) / ( σ/√(n)) < ( 71 - 70 ) / ( 4 / √(64) )
P ( X < 71 ) = P ( Z < 2 )
P ( X̅ < 71 ) = 0.9772
No, it would not be unusual because more than 5% of all such samples have means less than 71.