A vending machine dispenses coffee into an eight-ounce cup. The amount of coffee dispensed into the cup is normally distributed with a standard deviation of .03 ounce. You can allow the cup to OVERFILL 1% of the time. What amount should you set as the mean amount of coffee to be dispensed?
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To determine the mean amount of coffee to be dispensed, we need to find the value that corresponds to the 1st percentile of the normal distribution. This is the value at which 1% of the coffee cups would overfill.
Step 1: Find the Z-score corresponding to the 1st percentile
The 1st percentile corresponds to the cumulative probability of 0.01. We can find the Z-score using a Z-table or a statistical calculator. For a standard normal distribution, the Z-score corresponding to a cumulative probability of 0.01 is approximately -2.33.
Step 2: Apply the Z-score formula
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Value in the distribution
μ = Mean
σ = Standard deviation
In this case, we know Z = -2.33, σ = 0.03, and we want to solve for μ.
Step 3: Solve for μ
Rearranging the formula, we get:
μ = X - (Z * σ)
Given that we want to find the mean amount that corresponds to 1% overfilling, we can set X = 8.03 ounces (8 ounces + 0.03 ounces).
Substituting the known values into the formula:
μ = 8.03 - (-2.33 * 0.03)
μ = 8.03 + 0.0699
μ ≈ 8.0999
Therefore, you should set the mean amount of coffee to be dispensed as approximately 8.0999 ounces to allow the cup to overfill 1% of the time.