Question

A simple random sample of size n=64 is obtained from a population with μ=88 and σ=24.

​(a) Describe the sampling distribution of x. is it 1,2,3,4, or 5?

1. The distribution is skewed right.

2. The distribution is approximately normal.

3. The distribution is uniform.

4. The distribution is skewed left.

5. The shape of the distribution is unknown.   

Find the mean and standard deviation of the sampling distribution of x.

Ux=

Ox=

​(b) What is P (x>91.75) ? Round to four decimal places as needed.

​(c) What is P (x≤81.55) ? Round to four decimal places as needed.

​(d) What is P (85<x<93.7) ? Round to four decimal places as needed.

Answer 1

Solution :

Given that,

mean = = 88

standard deviation = = 24

n = 64

a) 2. The distribution is approximately normal.

= = 88

= / n = 24 / 64 = 3

b) P( > 91.75) = 1 - P( < 91.75)

= 1 - P[( - ) / < (91.75 - 88) / 3]

= 1 - P(z < 1.25)

Using z table,    

= 1 - 0.8944

= 0.1056

c) P( 81.55 ) = P(( - ) / (81.55 - 88) / 3 )

= P(z -2.15)

Using z table

= 0.0158

d) P( 85 < < 93.7 )  

= P[(85 - 88) /3 < ( - ) / < (93.7 - 88) / 3)]

= P(-1.00 < Z < 1.90)

= P(Z < 1.90) - P(Z < -1.00)

Using z table,  

= 0.9713 - 0.1587

= 0.8126