Question

Given a normal distribution with μ = 100 and σ=10, complete parts​ (a) through​ (d).

a. What is the probability that X > 95?

The probability that X > 95 is ___.

​(Round to four decimal places as​ needed.)

b. What is the probability that X < 75​?

The probability that X < 75 is ___.

​(Round to four decimal places as​ needed.)

c. What is the probability that X < 85 or X > 110​?

The probability that X < 85 or X > 110 is___.

​(Round to four decimal places as​ needed.)

d. 90​% of the values are between what two​ X-values (symmetrically distributed around the​ mean)?

90​% of the values are greater than ____ and less than ____.

​(Round to four decimal places as​ needed.)

Answer 1

Answer:

Given Data

a) What is the probability that X > 95?

The probability that X > 95 is

X ~ N ( µ = 100 , σ = 10 )
P ( X > 95 ) = 1 - P ( X < 95 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 95 - 100 ) / 10
Z = - 0.5
P ( ( X - µ ) / σ ) > ( 95 - 100 ) / 10 )
P ( Z > -0.5 )
P ( X > 95 ) = 1 - P ( Z < -0.5 )
P ( X > 95 ) = 1 - 0.3085
P ( X > 95 ) = 0.6915

b) What is the probability that X < 75​?

The probability that X <

X ~ N ( µ = 100 , σ = 10 )
P ( X < 75 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 75 - 100 ) / 10
Z = - 2.5
P ( ( X - µ ) / σ ) < ( 85 - 100 ) / 10 )
P ( X < 75 ) = P ( Z < -2.5 )
P ( X < 75 ) = 0.00621

c)  What is the probability that X < 85 or X > 110​?

The probability that X < 85 or X > 110 is__

X ~ N ( µ = 100 , σ = 10 )
P ( 85 < X < 110 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 85 - 100 ) / 10
Z = -1.5
Z = ( 110 - 100 ) / 10
Z = 1.0
P ( - 1.5 < Z < 1.0 )
P ( 85 < X < 110 ) = P ( Z < 1.0 ) - P ( Z < -1.5 )
P ( 85 < X < 110 ) = 0.158655 - 0.066807
P ( 85 < X < 110 ) = 0.091848

Required probability = 1 - 0.091848

= 0. 908152

d)  90​% of the values are between what two​ X-values (symmetrically distributed around the​ mean)?

90​% of the values are greater than ____ and less than

X ~ N ( µ = 100 , σ = 10 )
P ( a < X < b ) = 0.9
Dividing the area 0.9 in two parts we get 0.9/2 = 0.45
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.45
Area above the mean is b = 0.5 + 0.45
Looking for the probability 0.05 in standard normal table to calculate Z score = -1.6449
Looking for the probability 0.95 in standard normal table to calculate Z score = 1.6449
Z = ( X - µ ) / σ
-1.6449 = ( X - 100 ) / 10
a = 83.55
1.6449 = ( X - 100 ) / 10
b = 116.45

P ( 83.55 < X < 116.45 ) = 0.90

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