QuestionA vending machine is designed to dispense a mean of 7.4 oz of coffee into an 8 -oz cup. If the standard deviation of the amount of coffee dispensed is 0.3 oz and the amount is normally distributed, find the percent of times the machine will dispense less than 7.64 oz .
Click here to view page 1 of the standard normal distribution table.
Click here to view page 2 of the standard normal distribution table.
The percentage of times the machine will dispense less than 7.64 oz is
(Type an integer or a decimal rounded to two decimal places as needed.)
Studdy Solution
STEP 1
What is this asking? What percentage of the time does a vending machine dispense less than **7.64 oz** of coffee, if the average is **7.4 oz** and the standard deviation is **0.3 oz**? Watch out! Don't forget to convert to a *z*-score before using the *z*-table!
STEP 2
1. Calculate the *z*-score.
2. Find the probability using the *z*-table.
3. Convert to a percentage.
STEP 3
The *z*-score tells us how many **standard deviations** a value is away from the **mean**.
The formula is:
where is the **value** we're interested in, is the **mean**, and is the **standard deviation**.
STEP 4
In our case, oz, oz, and oz. **Plug these values** into the formula:
STEP 5
**Simplify** the numerator:
STEP 6
**Divide** to find the *z*-score: So, **7.64 oz** is **0.8 standard deviations** above the mean.
STEP 7
Our *z*-score is **0.8**.
The *z*-table tells us the probability of getting a value *less than* a given *z*-score.
Look up in the *z*-table.
STEP 8
The corresponding probability for is **0.7881**.
This means there's a **0.7881** probability that the vending machine dispenses less than **7.64 oz**.
STEP 9
To convert the probability to a percentage, **multiply** by 100:
STEP 10
So, the percentage is **78.81%**.
STEP 11
The vending machine will dispense less than 7.64 oz approximately **78.81%** of the time.
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