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 0.07 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? Click to view page 1 of the table. Click to view page 2 of the table. ounces (Round to two decimal places as needed.)

Step 1 ：The problem is asking for the mean amount of coffee to be dispensed such that the cup overfills only 1% of the time. This is a problem of normal distribution. We know that the standard deviation is 0.07 ounces. We also know that the cup's capacity is 8 ounces. We need to find the mean such that 99% of the time, the amount of coffee dispensed is less than or equal to 8 ounces.

Step 2 ：We can use the Z-score formula to solve this problem. The Z-score is a measure of how many standard deviations an element is from the mean. In a normal distribution, about 99% of the data will fall within ±2.33 standard deviations of the mean.

Step 3 ：So, we can set up the following equation to find the mean (μ): \(8 = μ + 2.33(0.07)\)

Step 4 ：We can solve this equation to find the mean. The mean amount of coffee to be dispensed should be set to \(\boxed{7.84}\) ounces.