Question

(1 point) To estimate the mean height μ of male students on your campus, you will measure an SRS of students. Heights of people of the same sex and similar ages are close to Normal. You know from government data that the standard deviation of the heights of young men is about 2.8 inches. Suppose that (unknown to you) the mean height of all male students is 70 inches.

(a) If you choose one student at random, what is the probability that he is between 69 and 73 inches tall? (

b) You measure 49 students. What is the standard deviation of the sampling distribution of their average height x⎯⎯⎯? (c) What is the probability that the mean height of your sample is between 69 and 73 inches?

Answer 1

a)

X ~ N ( µ = 70 , σ = 2.8 )
P ( 69 < X < 73 ) = ?
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 69 - 70 ) / 2.8
Z = -0.36
Z = ( 73 - 70 ) / 2.8
Z = 1.07
P ( -0.36 < Z < 1.07 )
P ( 69 < X < 73 ) = P ( Z < 1.07 ) - P ( Z < -0.36 )
P ( 69 < X < 73 ) = 0.8577 - 0.3594
P ( 69 < X < 73 ) = 0.4983

b)

The standard deviation of the sampling distribution of = σ / sqrt(n)

= 2.8 / sqrt(49)

= 0.4

c)

P ( 69 < X̅ < 73 ) = ?
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 69 - 70 ) / ( 2.8 / √(49))
Z = -2.5
Z = ( 73 - 70 ) / ( 2.8 / √(49))
Z = 7.5
P ( -2.5 < Z < 7.5 )
P ( 69 < X̅ < 73 ) = P ( Z < 7.5 ) - P ( Z < -2.5 )
P ( 69 < X̅ < 73 ) = 1 - 0.0062
P ( 69 < X̅ < 73 ) = 0.9938