XYZ distribution center receives 1,000 deliveries each day.Arrivals are on time if they are within plus or minus 10 minutes of the targeted arrival.In order to have only 3.4 late or early arrivals per million deliveries,what would the standard deviation of arrival times need to be?
A)3.33 minutes
B)0.83 minutes
C)1.67 minutes
D)6.67 minutes

Question

Quizplus · Accepted Answer

First, we need to calculate the number of standard deviations that correspond to 3.4 deviations from the mean in a normal distribution. Using the formula for z-score:

z = (x - μ) / σ

where z is the number of standard deviations from the mean, x is the actual value, μ is the mean, and σ is the standard deviation.

For 3.4 deviations from the mean in a normal distribution, we can use a z-table or a calculator to find that the corresponding z-score is 3.0902.

Next, we need to find the standard deviation that makes it such that only 3.4 deliveries are late/early out of one million deliveries. We know that 1,000 deliveries are made each day, so the total number of deliveries in one year is:

1,000 * 365 = 365,000

To find the standard deviation, we can use the formula:

σ = (x - μ) / z

where x and μ are the targeted arrival time and z is the z-score corresponding to the desired number of deviations from the mean.

Let's assume that the targeted arrival time is exactly on time (no early or late arrivals). We can convert our desired number of late/early arrivals to a decimal:

3.4 / 1,000,000 = 0.0000034

Now we can plug in our values:

σ = (10 - 0) / 3.0902 * sqrt(365,000 * 0.0000034)

σ ≈ 1.67 minutes

Therefore, the standard deviation of arrival times needs to be approximately 1.67 minutes for there to be only 3.4 late or early arrivals per million deliveries. The answer is C.

XYZ Distribution Center Receives 1,000 Deliveries Each Day