Questions: A vending machine dispenses coffee into a sixteen-ounce cup. The amount of coffee dispensed into the cup is normally distributed with a standard deviation of 0.03 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? ounces (Round to two decimal places as needed.)

A vending machine dispenses coffee into a sixteen-ounce cup. The amount of coffee dispensed into the cup is normally distributed with a standard deviation of 0.03 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?

ounces (Round to two decimal places as needed.)

Transcript text: A vending machine dispenses coffee into a sixteen-ounce cup. The amount of coffee dispensed into the cup is normally distributed with a standard deviation of 0.03 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? Click to view page 1 of the table. Click to view page 2 of the table. $\square$ ounces (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Identify the Problem

We need to determine the mean amount of coffee to be dispensed into a 16-ounce cup such that the cup overfills only $2\%$ of the time. The coffee amount is normally distributed with a standard deviation of $0.03$ ounces.

Step 2: Determine the Z-score

To find the mean, we first need the Z-score corresponding to the $98\%$ percentile (since $2\%$ overfill is allowed). The Z-score for the $98\%$ percentile is approximately $2.0537$.

Step 3: Calculate the Mean Amount

Using the Z-score formula for a normal distribution: \[ \text{mean} = \text{cup capacity} - (Z \times \text{standard deviation}) \] Substitute the known values: \[ \text{mean} = 16 - (2.0537 \times 0.03) \] \[ \text{mean} \approx 16 - 0.0616 = 15.9384 \]

Step 4: Round the Mean Amount

Round the mean amount to two decimal places: \[ \text{mean} \approx 15.94 \]