Question

A simple random sample of size

nequals=8181

is obtained from a population with

mu equals 73μ=73

and

sigma equals 18σ=18.

​(a) Describe the sampling distribution of

x overbarx.

​(b) What is

Upper P left parenthesis x overbar greater than 75.9 right parenthesisP x>75.9​?

​(c) What is

Upper P left parenthesis x overbar less than or equals 68.8 right parenthesisP x≤68.8​?

​(d) What is

Upper P left parenthesis 71 less than x overbar less than 77.8 right parenthesisP 71<x<77.8​?

Answer 1

Using central limit theorem,

P( < x) = P( Z < x - / ( / sqrt(n) ) )

a)

P( > 75.9) = P( Z > 75.9 - 73 / (18 / sqrt(81) ) )

= P( Z > 1.45)

= 1 - P( Z < 1.45)

= 1 - 0.9265 (Probability calculated from Z table)

= 0.0735

b)

P( <= 68.8) = P( Z <= 68.8 - 73 / ( 18 / sqrt(81) ) )

= P( Z <= -2.1)

= 1 - P( Z < 2.1)

= 1 - 0.9821 (Probability calculated from Z table)

= 0.0179

c)

P(71 < < 77.8) = P( < 77.8) - P( < 71)

= P( Z < 77.8 - 73 / ( 18 / sqrt(81) ) ) - P( Z < 71 - 73 / ( 18 / sqrt(81) ) )

= P( Z < 2.4) - P( Z < -1)

= P( Z < 2.4) - ( 1 - P (Z < 1) )

= 0.9918 - ( 1 - 0.8413 ) (Probability calculated from Z table)

= 0.8331