To determine the mean amount of coffee to be dispensed, we need to find the value that corresponds to the top 1% of the distribution (the amount that can allow the cup to overfill 1% of the time).
First, we need to understand the concept of the standard normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1. We can convert our problem into a standard normal distribution by using the z-score formula:
z = (x - μ) / σ
where z is the z-score, x is the value, μ is the mean, and σ is the standard deviation.
In our case, we want to find the z-score that corresponds to the top 1%. Since the normal distribution is symmetric, we can find the value that corresponds to the bottom 1% and then use that to find the top 1%.
To find the z-score that corresponds to the bottom 1%, we need to find the z-score that corresponds to an area of 0.01 in the left tail of the standard normal distribution. We can use a table or statistical software to find this value.
Using a standard normal distribution table or statistical software, we find that the z-score corresponding to an area of 0.01 in the left tail is approximately -2.33.
Now, we can use the z-score formula to find the value that corresponds to this z-score in our original distribution:
z = (x - μ) / σ
-2.33 = (x - μ) / 0.03
Rearranging the equation, we can solve for the mean amount of coffee, μ:
μ = x - 2.33 * 0.03
Since we want to dispense 8 ounces of coffee on average, we can substitute x with 8:
μ = 8 - 2.33 * 0.03
Calculating this gives us:
μ ≈ 7.93
Therefore, you should set the mean amount of coffee to be dispensed at approximately 7.93 ounces in order to allow the cup to overfill 1% of the time.