A vending machine is designed to dispense a mean of 7.6 oz of coffee into an 8-oz cup. If the standard deviation of the amount of coffee dispensed is 0.4 oz and the amount is normally distributed, find the percent of times the machine will
a) dispense less than 7.0 oz.
b) result in the cup overflowing (therefore dispense more than 8 oz).
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores. Multiply the proportion by 100 to get the percentage.
To find the percent of times the machine will dispense less than 7.0 oz and result in the cup overflowing (dispensing more than 8 oz), we will use the Z-score formula and the Z-table.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
- X is the value we want to find the probability for,
- μ is the mean,
- σ is the standard deviation.
Let's calculate the probabilities step-by-step:
a) Probability of dispensing less than 7.0 oz:
To find this probability, we need to find the Z-score for X = 7.0 oz, and then use the Z-table to find the corresponding area under the normal curve.
Z = (7.0 - 7.6) / 0.4
Z = -1.5
Using the Z-table, we find that the area to the left of Z = -1.5 is approximately 0.0668.
So, the machine will dispense less than 7.0 oz approximately 6.68% of the time.
b) Probability of cup overflowing (dispensing more than 8.0 oz):
Similarly, we need to find the Z-score for X = 8.0 oz, and then use the Z-table to find the corresponding area under the normal curve.
Z = (8.0 - 7.6) / 0.4
Z = 1.0
Using the Z-table, we find that the area to the left of Z = 1.0 is approximately 0.8413.
To find the probability of overflowing, we subtract this area from 1 (since we're looking for the area to the right of 8.0 oz):
P(X > 8.0) = 1 - 0.8413
P(X > 8.0) = 0.1587
So, the machine will result in the cup overflowing (dispensing more than 8.0 oz) approximately 15.87% of the time.
To find the percentage of times the vending machine will dispense a certain amount of coffee, we need to use the concept of Z-scores and the standard normal distribution.
a) To calculate the percent of times the machine will dispense less than 7.0 oz of coffee, we need to find the Z-score for this value and then use a Z-table or a calculator with the capability to find the cumulative probability (area) under the normal curve.
Step 1: Calculate the Z-score
Z = (X - μ) / σ
where X is the value we want to find the percentage for (7.0 oz), μ is the mean (7.6 oz), and σ is the standard deviation (0.4 oz).
Z = (7.0 - 7.6) / 0.4
Z = -1.5
Step 2: Find the cumulative probability
Using a Z-table or a calculator, we can find the cumulative probability associated with the Z-score of -1.5. This will give us the percentage of times the machine will dispense less than 7.0 oz of coffee.
b) Similarly, to find the percentage of times the machine will result in the cup overflowing (dispensing more than 8 oz of coffee), we follow the same steps.
Step 1: Calculate the Z-score
Z = (X - μ) / σ
where X is the value we want to find the percentage for (8.0 oz), μ is the mean (7.6 oz), and σ is the standard deviation (0.4 oz).
Z = (8.0 - 7.6) / 0.4
Z = 1
Step 2: Find the cumulative probability
Using a Z-table or a calculator, we can find the cumulative probability associated with the Z-score of 1. This will give us the percentage of times the machine will dispense more than 8.0 oz of coffee.
Note: The standard normal distribution table or calculator will give the probability (area) to the left or right of the Z-score. To find the answer to "a," we need to find the area to the left of -1.5, and for "b," we need to find the area to the right of 1.