A vending machine dispenses coffee into a sixteen ounce cup. The amount of coffee dispensed into the cup is normally distributed with a standard deviation of 0.03 ounces. You can allow the cup to overfill 10% of the time. What amount (in ounces) should you set as the mean amount of coffee to be dispensed?
To determine the mean amount of coffee to be dispensed, we need to consider the desired overfill rate. In this case, we want the cup to overfill 10% of the time.
First, let's start with the standard normal distribution, which has a mean of 0 and a standard deviation of 1. We need to find the z-score that corresponds to a cumulative probability of 0.10 (10%). Using a standard normal table or a calculator, we find that the z-score is approximately 1.28.
Since we want to account for the cup overfilling, we need to subtract this z-score from the desired amount of coffee in ounces. Let's denote the mean amount of coffee as μ.
We can use the formula for z-score:
z = (x - μ) / σ
Here, x represents the desired amount of coffee in ounces, and σ is the standard deviation (0.03 ounces).
Plugging in the values we know:
1.28 = (16 - μ) / 0.03
Now, let's solve the equation for μ:
1.28 * 0.03 = 16 - μ
0.0384 = 16 - μ
μ = 16 - 0.0384
μ ≈ 15.9616
Therefore, you should set the mean amount of coffee to be dispensed at approximately 15.9616 ounces.