Question

Suppose that the average number of Facebook friends users have is normally distributed with a mean of 133 and a standard deviation of about 55.
Assume twelve individuals are randomly chosen. Answer the following questions. Round all answers to 4 decimal places where possible.

a. For the group of 12, find the probability that the average number of friends is more than 117.

b. Find the first quartile for the average number of Facebook friends.  [SAB1

Answer 1

Part a)

X ~ N ( µ = 133 , σ = 55 )
P ( X > 117 ) = 1 - P ( X < 117 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 117 - 133 ) / ( 55 / √ ( 12 ) )
Z = -1.0077
P ( ( X - µ ) / ( σ / √ (n)) > ( 117 - 133 ) / ( 55 / √(12) )
P ( Z > -1.01 )
P ( X̅ > 117 ) = 1 - P ( Z < -1.01 )
P ( X̅ > 117 ) = 1 - 0.1568
P ( X̅ > 117 ) = 0.8432

Part b)

X ~ N ( µ = 133 , σ = 55 )
P ( X < x ) = 25% = 0.25
To find the value of x
Looking for the probability 0.25 in standard normal table to calculate Z score = -0.6745
Z = ( X - µ ) / ( σ / √(n) )
-0.6745 = ( X - 133 ) / ( 55/√(12) )
X = 122.2909
P ( X < 122.2909 ) = 0.25