Question

In: Statistics and Probability

6) Given a variable with the following population parameters: Mean = 20 Variance = 16 a)...

6) Given a variable with the following population parameters: Mean = 20 Variance = 16
a) What is the probability of obtaining a score greater than 23?
b) What is the probability of obtaining a score greater than 16?
c) What is the probability of obtaining a score less than 10?
d) What is the probability of obtaining a score greater than 18 and less than 21? e) What is the probability of obtaining a score greater than 21 and less than 27? f) What is the score at which 75% of the data falls at or below?
g) What is the score at which 22% of the data falls at or above?
h) Within what two scores do 95% of the scores fall (i.e. symmetrically)?

Solutions

Expert Solution

a)

X ~ N ( µ = 20 , σ = 4 )
P ( X > 23 ) = 1 - P ( X < 23 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 23 - 20 ) / 4
Z = 0.75
P ( ( X - µ ) / σ ) > ( 23 - 20 ) / 4 )
P ( Z > 0.75 )
P ( X > 23 ) = 1 - P ( Z < 0.75 )
P ( X > 23 ) = 1 - 0.7734
P ( X > 23 ) = 0.2266

b)

P ( X > 16 ) = 1 - P ( X < 16 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 16 - 20 ) / 4
Z = -1
P ( ( X - µ ) / σ ) > ( 16 - 20 ) / 4 )
P ( Z > -1 )
P ( X > 16 ) = 1 - P ( Z < -1 )
P ( X > 16 ) = 1 - 0.1587
P ( X > 16 ) = 0.8413

c)

P ( X < 10 ) = ?
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 10 - 20 ) / 4
Z = -2.5
P ( ( X - µ ) / σ ) < ( 10 - 20 ) / 4 )
P ( X < 10 ) = P ( Z < -2.5 )
P ( X < 10 ) = 0.0062

d)

P(X > 18 and less than 21) = P(18 < X < 21)

Standardizing the value   
Z = ( X - µ ) / σ  
Z = ( 18 - 20 ) / 4  
Z = -0.5  
Z = ( 21 - 20 ) / 4  
Z = 0.25  
P ( -0.5 < Z < 0.25 )   
P ( 18 < X < 21 ) = P ( Z < 0.25 ) - P ( Z < -0.5 )   
P ( 18 < X < 21 ) = 0.5987 - 0.3085  
P ( 18 < X < 21 ) = 0.2902  

e)

P ( 21 < X < 27 ) = ?
Standardizing the value   
Z = ( X - µ ) / σ  
Z = ( 21 - 20 ) / 4  
Z = 0.25  
Z = ( 27 - 20 ) / 4  
Z = 1.75  
P ( 0.25 < Z < 1.75 )   
P ( 21 < X < 27 ) = P ( Z < 1.75 ) - P ( Z < 0.25 )   
P ( 21 < X < 27 ) = 0.9599 - 0.5987  
P ( 21 < X < 27 ) = 0.3612  

f)

P ( X < x ) = 75% = 0.75
To find the value of x
Looking for the probability 0.75 in standard normal table to calculate Z score = 0.6745
Z = ( X - µ ) / σ
0.6745 = ( X - 20 ) / 4
X = 22.70

g)

P ( X > x ) = 1 - P ( X < x ) = 1 - 0.22 = 0.78
To find the value of x
Looking for the probability 0.78 in standard normal table to calculate Z score = 0.7722
Z = ( X - µ ) / σ
0.7722 = ( X - 20 ) / 4
X = 23.09

h)

P ( a < X < b ) = 0.95
Dividing the area 0.95 in two parts we get 0.95/2 = 0.475
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.475
Area above the mean is b = 0.5 + 0.475
Looking for the probability 0.025 in standard normal table to calculate Z score = -1.96
Looking for the probability 0.975 in standard normal table to calculate Z score = 1.96
Z = ( X - µ ) / σ
-1.96 = ( X - 20 ) / 4
a = 12.16
1.96 = ( X - 20 ) / 4
b = 27.84


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