Question

8 for the 40 patients seen each day by a doctors office, the table shows the distribution of time, in minutes, that it takes to complete the checkout process.
checkout time (minutes) | 1 | 3 | 5 | 10
number of patients | 12 | 15 | 10 | 3
after the checkout desk is moved during construction, the checkout process increases by 2 minutes due to the additional walk from the exam room to the checkout desk. let ( w ) = the time, in minutes, that it takes a randomly selected patient to complete the checkout process during construction. what is the expected value of ( w )?
a) 1.425 minutes
b) 3 minutes
c) 3.425 minutes
d) 5.425 minutes
e) 21 minutes

Explanation:

Step1: Calculate original expected time

First, find the expected value of the original checkout time \(E(X)\):
\[
E(X) = \frac{(1 \times 12) + (3 \times 15) + (5 \times 10) + (10 \times 3)}{40}
\]
\[
E(X) = \frac{12 + 45 + 50 + 30}{40} = \frac{137}{40} = 3.425
\]

Step2: Adjust for 2-minute increase

Since \(W = X + 2\), use linearity of expectation:
\[
E(W) = E(X) + 2
\]
\[
E(W) = 3.425 + 2 = 5.425
\]

Answer:

D) 5.425 minutes