Question

4.44 Heights of 10 year olds: Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 55 inches and standard deviation 6 inches.


(a) What is the probability that a randomly chosen 10 year old is shorter than 48 inches?
____ (please round to four decimal places)


(b) What is the probability that a randomly chosen 10 year old is between 60 and 65 inches?
(please round to four decimal places)


(c) If the tallest 10% of the class is considered "very tall", what is the height cutoff for "very tall"?
inches ____ (please round to two decimal places)


(d) The height requirement for Batman the Ride at Six Flags Magic Mountain is 54 inches. What proportion of 10 year olds cannot go on this ride?
____ (please round to four decimal places)

Answer 1

a)

X ~ N ( µ = 55 , σ = 6 )

We convert this to standard normal as

P ( X < x ) = P ( Z < ( X - µ ) / σ )

P ( X < 48 ) = P ( Z < 48 - 55 ) / 6 )

= P ( Z < -1.17 )

P ( X < 48 ) = 0.1210 (From Z table)

b)

Given :- = 55 , = 6 )

We convet this to Standard Normal as

P(X < x) = P( Z < ( X - ) / )

P ( 60 < X < 65 ) = P ( Z < ( 65 - 55 ) / 6 ) - P ( Z < ( 60 - 55 ) / 6 )

= P ( Z < 1.67) - P ( Z < 0.83 )

= 0.9525 - 0.7967 (From Z table)

= 0.1558

c)

X ~ N ( µ = 55 , σ = 6 )

P ( X > x ) = 0.10

P ( X < x ) = 1 - 0.1 = 0.9

Z = ( X - µ ) / σ

P(X < x) = P(Z < ( X - µ ) / σ)

To find the value of x

Looking for the probability 0.9 in standard normal table to calculate critical value Z = 1.2816

( X - µ ) / σ = 1.2816

( X - 55 ) / 6 = 1.2816

X = 62.69

d)

X ~ N ( µ = 55 , σ = 6 )

We convert this to standard normal as

P ( X < x ) = P ( Z < ( X - µ ) / σ )

P ( ( X < 54 ) = P ( Z < 54 - 55 ) / 6 )

= P ( Z < -0.17 )

P ( X < 54 ) = 0.4325

= 43.25%