Respuesta :
Answer:
a) 36.9% of customers receive the service for half-price
b) The automotive center should make 21.28 minutes, the guaranteed time limit.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 19 minutes
Standard Deviation, σ = 3 minutes.
We are given that the distribution of time required is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
a) P(service will take no longer than 20 minutes)
P(x < 20)
[tex]P( x < 20) = P( z < \displaystyle\frac{20 - 19}{3}) = P(z < 0.3334)[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(x < 20) = 0.631 = 63.1\%[/tex]
If it does take longer than 20 minutes, the customer will receive the service for half-price.
Customers receive the service for half-price
= [tex]100\%-63.1\% = 36.9\%[/tex]
b) We have to find the value of x such that the probability is 0.77.
P(X < x)
[tex]P( X < x) = P( z < \displaystyle\frac{x - 19}{3})=0.77[/tex]
Calculation the value from standard normal z table, we have,
[tex]P(z < 0.739) = 0.77[/tex]
[tex]\displaystyle\frac{x - 19}{3} = 0.739\\x = 21.28[/tex]
Hence, the automotive center should make 21.28 minutes, the guaranteed time limit.
