A vending machine dispenses coffee into a twelve-ounce cup. The amount of coffee dispensed into the cup is normally distributed with a standard deviation of 0.05 ounce. You can allow the cup to overfill 4% of the time. What amount should you set as the mean amount of coffee to be dispensed
Find one-tail z-score from tables
Z(0.96)=(12-μ)/0.05
Solve for μ
To determine the mean amount of coffee to be dispensed, we need to find the amount of coffee that corresponds to the 96th percentile of the normal distribution.
Since the cup can overfill 4% of the time, we want to find the value that corresponds to the upper 96th percentile of the distribution.
Step 1: Find the z-score corresponding to the upper 96th percentile.
Using a standard normal distribution table or a calculator, we find that the z-score corresponding to the upper 96th percentile is approximately 1.75.
Step 2: Calculate the mean.
We know that the mean of a normal distribution corresponds to the center or peak of the distribution. In this case, we want to set the mean amount of coffee to be dispensed such that 96% of the time, the cup is not overfilled.
Using the z-score formula: z = (X - μ) / σ,
where z is the z-score, X is the desired amount of coffee, μ is the mean, and σ is the standard deviation, we can rearrange the formula to solve for the mean:
μ = X - (z * σ)
Substituting the values we have:
μ = X - (1.75 * 0.05)
Step 3: Solve for the mean.
As we want to dispense coffee into a twelve-ounce cup, we substitute X = 12 into the equation:
μ = 12 - (1.75 * 0.05)
μ ≈ 12 - 0.0875
μ ≈ 11.9125
Therefore, you should set the mean amount of coffee to be dispensed as approximately 11.9125 ounces.
To determine the mean amount of coffee to be dispensed, we need to find the value that corresponds to the top 4% of the normal distribution.
Step 1: Find the Z-score corresponding to the top 4% of the distribution.
Since we want to find the top 4%, which means an area of 0.04 under the curve, we can use a standard normal distribution table or a calculator to find the Z-score.
Using a standard normal distribution table or a calculator, you can find that the Z-score corresponding to the top 4% is approximately 1.75.
Step 2: Calculate the mean amount of coffee.
Now, we need to calculate the mean amount of coffee using the Z-score formula:
Z = (X - μ) / σ
Where:
Z is the Z-score (1.75),
X is the amount of coffee (unknown),
μ is the mean amount of coffee to be dispensed, and
σ is the standard deviation (0.05 ounce).
Rearranging the formula to solve for μ:
μ = X - (Z * σ)
Substituting the known values:
μ = X - (1.75 * 0.05)
Simplifying:
μ = X - 0.0875
Therefore, the mean amount of coffee to be dispensed should be X - 0.0875 ounces.