Question

A government sample survey plans to measure the LDL (bad) cholesterol level of an SRS of men aged 20 to 34.
Suppose that in fact the LDL cholesterol level of all men aged 20 to 34 follows the Normal distribution with mean

μ = 119 milligrams per deciliter (mg/dL) and standard deviation   σ = 25 mg/dL. Use Table A for the following questions, where necessary.

(a) What is the probability that x takes a value between 116 and 122 mg/dL? This is the probability that x estimates μ within ±3 mg/dL. (Round your answer to three decimal places.)

(b) Choose an SRS of 1000 men from this population. Now what is the probability that x falls within ±3 mg/dL of μ? The larger sample is much more likely to give an accurate estimate of μ. (Use Table A and round your answer to 3 decimal places.)

Answer 1

a)

Given :- = 119 , = 25 )
We convet this to Standard Normal as
P(X < x) = P( Z < ( X - ) / )
P ( 116 < X < 122 ) = P ( Z < ( 122 - 119 ) / 25 ) - P ( Z < ( 116 - 119 ) / 25 )
= P ( Z < 0.12) - P ( Z < -0.12 )
= 0.5478 - 0.4522 (From Z table)
= 0.096

b)

X ~ N ( µ = 119 , σ = 25 )
P ( 116 < X < 122 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 116 - 119 ) / ( 25 / √(1000))
Z = -3.79
Z = ( 122 - 119 ) / ( 25 / √(1000))
Z = 3.79
P ( 116 < X̅ < 122 ) = P ( Z < 3.79 ) - P ( Z < -3.79 )
P ( 116 < X̅ < 122 ) = 0.9999 - 0.0001 (From Z table)
P ( 116 < X̅ < 122 ) = 1.000