Question

7. The Edwards’s Theater chain has studied its movie customers to determine how much money they spend on concessions. The study revealed that the spending distribution is approximately normally distributed with a mean of $4.11 and a standard deviation of $1.37.

a. If a customer is randomly selected, find the probability that they spent less than $3 on concessions.

b. If 10 customers are randomly selected, find the probability that they spent less than $3 on concessions. (This is a sample, so make sure you use the central limit theorem to come up with your standard deviation).

Answer 1

a)

X ~ N ( µ = 4.11 , σ = 1.37 )
We convert this to standard normal as
P ( X < x ) = P ( Z < ( X - µ ) / σ )
P ( ( X < 3 ) = P ( Z < 3 - 4.11 ) / 1.37 )
= P ( Z < -0.81 )
P ( X < 3 ) = 0.2090 (From Z table)

b)

X ~ N ( µ = 4.11 , σ = 1.37 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 3 - 4.11 ) / ( 1.37 / √10 )
Z = -2.56
P ( ( X - µ ) / ( σ/√(n)) = ( 3 - 4.11 ) / ( 1.37 / √(10) )
P ( X < 3 ) = P ( Z < -2.56 )
= 0.0052 (From Z table)