A vending machine dispenses coffee into a twelve-ounce cup. The amount of coffee dispensed in to the cup is normally distributed with a standard deviation of .08 ounce. You can allow the cup to overfill 2% of the time. What amount should you set as the mean amount of coffee to be dispensed?
12.23
To determine the mean amount of coffee to be dispensed, we need to find the value of the mean that results in only 2% of the cups being overfilled.
First, we need to identify the z-score associated with a 2% overfill rate. The z-score represents the number of standard deviations away from the mean a particular value is in a normal distribution.
To find the z-score, we can use a standard normal distribution table or a statistical calculator. In this case, since we know the cumulative area is 2% (0.02) on one tail of the distribution, we need to find the corresponding z-score.
Using a standard normal distribution table, we can find that the z-score associated with a cumulative area of 0.02 is approximately -2.05.
Next, we use the z-score formula to find the mean:
z = (x - μ) / σ
Where:
- z is the z-score (-2.05 in this case)
- x is the value of the variable (12 ounces, the size of the cup)
- μ is the mean we want to find
- σ is the standard deviation (0.08 ounces)
Rearranging the formula, we can solve for the mean:
-2.05 = (12 - μ) / 0.08
Now, let's solve for μ:
-2.05 * 0.08 = 12 - μ
-0.164 = 12 - μ
μ = 12 - (-0.164)
μ ≈ 12.164
So, you should set the mean amount of coffee to be dispensed at approximately 12.164 ounces.