Question

Heights of 10 year olds, regardless of gender, closely follow a normal distribution with mean 52 inches and standard deviation 8 inches.

(a) What is the probability that a randomly chosen 10 year old is shorter than 49 inches?
(b) What is the probability that a randomly chosen 10 year old is between 45 and 63 inches?
(c) If the tallest 14% of the class is considered very tall, what is the height cutoff for very tall? inches
(d) Find a value k so that 25% of 10 year olds are within k inches of 52 inches tall. k=
(e) The height requirement for Batman the Ride at Six Flags Magic Mountain is 54 inches. (Children who are too tall cannot safely ride.) What percent of 10 year olds cannot go on this ride?

Answer 1

This is a normal distribution question with

a) P(x < 49.0)=?
The z-score at x = 49.0 is,

z = -0.375
This implies that
P(x < 49.0) = P(z < -0.375) = 0.3538
b) P(45.0 < x < 63.0)=?

This implies that
P(45.0 < x < 63.0) = P(-0.875 < z < 1.375) = P(Z < 1.375) - P(Z < -0.875)
P(45.0 < x < 63.0) = 0.9154 - 0.1908
P(45.0 < x < 63.0) = 0.7246
c) Given in the question
P(X < x) = 0.86
This implies that
P(Z < 1.0803) = 0.86
With the help of formula for z, we can say that

d) Given in the question
P(X < x) = 0.625 (25% of 10 years old are within 52 inches, that means 12.5% above 52 and 12.5% below 52)
This implies that
P(Z < 0.3186) = 0.625
With the help of formula for z, we can say that

e) P(x < 54.0)=?
The z-score at x = 54.0 is,

z = 0.25
This implies that
P(x < 54.0) = P(z < 0.25) = 0.5987
PS: you have to refer z score table to find the final probabilities.
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