Question

Given a variable with the following population parameters: Mean = 20 Variance = 16 Sample Size = 35

a) What is the probability of obtaining a mean greater the 23?

b) What is the probability of obtaining a mean less than 21?

c) What is the probability of obtaining a mean less than 18.2?

d) What is the probability of obtaining a mean greater than 19.5 and less than 21.2?

e) What is the probability of obtaining a mean greater than 20.5 and less than 21.5?

f) What is the sample mean at which 65% of the data falls at or below?

g) what is the sample mean at which 82% of the data falls at or above?

h) Within what two sample means do 90% of the means fall (i.e symmetrically)?

Answer 1

Part a)

X ~ N ( µ = 20 , σ = 4 )
P ( X > 23 ) = 1 - P ( X < 23 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 23 - 20 ) / ( 4 / √ ( 35 ) )
Z = 4.4371
P ( ( X - µ ) / ( σ / √ (n)) > ( 23 - 20 ) / ( 4 / √(35) )
P ( Z > 4.44 )
P ( X̅ > 23 ) = 1 - P ( Z < 4.44 )
P ( X̅ > 23 ) = 1 - 1
P ( X̅ > 23 ) = 0

Part b)

X ~ N ( µ = 20 , σ = 4 )
P ( X < 21 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 21 - 20 ) / ( 4 / √35 )
Z = 1.479
P ( ( X - µ ) / ( σ/√(n)) < ( 21 - 20 ) / ( 4 / √(35) )
P ( X < 21 ) = P ( Z < 1.48 )
P ( X̅ < 21 ) = 0.9304

Part c)

X ~ N ( µ = 20 , σ = 4 )
P ( X < 18.2 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 18.2 - 20 ) / ( 4 / √35 )
Z = -2.6622
P ( ( X - µ ) / ( σ/√(n)) < ( 18.2 - 20 ) / ( 4 / √(35) )
P ( X < 18.2 ) = P ( Z < -2.66 )
P ( X̅ < 18.2 ) = 0.0039

Part d)

X ~ N ( µ = 20 , σ = 4 )
P ( 19.5 < X < 21.2 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 19.5 - 20 ) / ( 4 / √(35))
Z = -0.7395
Z = ( 21.2 - 20 ) / ( 4 / √(35))
Z = 1.7748
P ( -0.74 < Z < 1.77 )
P ( 19.5 < X̅ < 21.2 ) = P ( Z < 1.77 ) - P ( Z < -0.74 )
P ( 19.5 < X̅ < 21.2 ) = 0.962 - 0.2298
P ( 19.5 < X̅ < 21.2 ) = 0.7322

Part e)

X ~ N ( µ = 20 , σ = 4 )
P ( 20.5 < X < 21.5 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 20.5 - 20 ) / ( 4 / √(35))
Z = 0.7395
Z = ( 21.5 - 20 ) / ( 4 / √(35))
Z = 2.2185
P ( 0.74 < Z < 2.22 )
P ( 20.5 < X̅ < 21.5 ) = P ( Z < 2.22 ) - P ( Z < 0.74 )
P ( 20.5 < X̅ < 21.5 ) = 0.9867 - 0.7702
P ( 20.5 < X̅ < 21.5 ) = 0.2165

part f)

X ~ N ( µ = 20 , σ = 4 )
P ( X ≤ x ) = 65% = 0.65
To find the value of x
Looking for the probability 0.65 in standard normal table to calculate Z score = 0.3853
Z = ( X - µ ) / ( σ / √(n) )
0.3853 = ( X - 20 ) / ( 4/√(35) )
X = 20.2605
P ( X ≤ 20.2605 ) = 0.65

part g)

X ~ N ( µ = 20 , σ = 4 )
P ( X ≥ x ) = 1 - P ( X < x ) = 1 - 0.82 = 0.18
To find the value of x
Looking for the probability 0.18 in standard normal table to calculate Z score = -0.9154
Z = ( X - µ ) / ( σ / √(n) )
-0.9154 = ( X - 20 ) / (4/√(35))
X = 19.3811
P ( X ≥ 19.3811 ) = 0.82

Part h)

X ~ N ( µ = 20 , σ = 4 )
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 / √(n) )
-1.6449 = ( X - 20 ) / ( 4/√(35) )
a = 18.8878
1.6449 = ( X - 20 ) / ( 4/√(35) )
b = 21.1122
P ( 18.8878 < X < 21.1122 ) = 0.9