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 0.07 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?
To determine the mean amount of coffee to be dispensed, we need to find the cutoff value of the normal distribution that corresponds to the top 2% of the distribution. This cutoff value will represent the maximum amount of coffee that will overfill the cup approximately 2% of the time.
Step 1: Determine the z-value corresponding to the 2% cutoff probability.
The z-value can be found using a standard normal distribution table or a statistical calculator. Since we're looking for the top 2% (the right-tail), we need to find the z-value for an area of 0.02 beyond the mean.
Using a standard normal table, we can find the z-value for a cumulative probability of 0.98 (1 - 0.02). The closest value is 2.05.
Step 2: Determine the cutoff value.
The cutoff value (X) can be found by multiplying the z-value by the standard deviation (σ) and adding it to the mean (μ):
X = μ + z * σ
In this case, the standard deviation (σ) is given as 0.07 ounce. We're solving for the mean (μ), so let's rewrite the equation as:
μ = X - z * σ
Substituting the values:
μ = 12 - (2.05 * 0.07)
μ = 12 - 0.1435
μ ≈ 11.8565
Therefore, the mean amount of coffee to be dispensed should be set at approximately 11.8565 ounces.