Given that x is the normally distributed random variable wit

Given that x is the normally distributed random variable with a mean of 60 and a standard deviation of 10, find the following probabilities:

A. P(x>60)
B. P(x < x < 72)
C. P(57 <x < 83)
D. P(65 < x < 82)
E. P(38 < x < 78)
F. P(X < 38)

Can you please tell me the calculator function and/or a formula I can use to get the answer, please? Thank you.

*NOTE* It is not a multiple choice question so all letters (A-F) must be answered. Thank you.

Solution

Given that x is the normally distributed random variable with

mean = 60 and

standard deviation = 10

find the following probabilities:

i) P(x>60)

For finding all these probabilities we need Z-score for x

Z = (x - µ) /

For x = 60

Z = (60 - 60) / 10 = 0

P(Z > 0) = 1 - P(Z <=0) = 1 - 0.5 = 0.5

ii) P(x < 72)

Z-score for x = 72,

Z = (72 - 60) / 10 = 1.2

P(Z < 1.2) = 0.8849

iii) P(57 <x < 83)

First find Z-score for x=57,

Z = (57 - 60) / 10 = -0.3

Z-score for x=83 ,

Z = (83 - 60) / 10 = 2.3

P(-0.3 < Z < 2.3) = P(Z < 2.3) - P(Z < -0.3) = 0.9893 - 0.3821 = 0.6072

iv) P(65 < x < 82)

Z-score for x=65,

Z = (65-60)/10 = 0.5

Z-score for x=82,

Z = (82-60)/10 = 2.2

P(0.5 < Z < 2.2) = P(Z < 2.2) - P(Z < 0.5) = 0.9861 - 0.6915 = 0.2946

v) P(38 < x < 78)

Z-score for x=38,

Z = (38 - 60)/10 = -2.2

Z-score for x=78,

Z = (78 - 60)/10 = 1.8

P(-2.2 < Z < 1.8) = P(Z <1.8) - P(Z <-2.2) = 0.9641 - 0.0139 = 0.9502

vi) P(X < 38)

Z-score for x = 38

Z = (38 - 60) / 10 = -2.2

P(Z < -2.2) = 0.0139

We can find all these probabilities by using EXCEL.

The syntax for that is,

=NORMSDIST(z)

where z is the Z-score value.