A Vending Machine is Designed to Dispense 8 Ounces of Coffee GMAT Problem Solving

Question: A vending machine is designed to dispense 8 ounces of coffee into a cup. After a test that recorded the number of ounces of coffee in each of 1000 cups dispensed by the vending machine, the 12 listed amounts, in ounces, were selected from the data above. If the 1000 recorded amounts have a mean of 8.1 ounces and a standard deviation of 0.3 ounces, how many of the 12 listed amounts are within 1.5 standard deviation of the mean?

1. Four
2. Six
3. Nine
4. Ten
5. Eleven

“A vending machine is designed to dispense 8 ounces of coffee”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “The Official Guide for Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:
Approach Solution 1:
If the Standard Deviation is 0.3 ounces, then 0.3 ounces represents 1 unit of standard deviation.

Similarly, 0.6 ounces represents 2 units of standard deviation,
0.15 ounces represents 0.5 units of standard deviation, and so on.

If the mean is 8.1 ounces, then we say that 8.4 ounces is 1 unit of standard deviation above the mean (since 8.1 + 0.3 = 8.4), and we say that 7.8 ounces is 1 unit of standard deviation below the mean (since 8.1 - 0.3 = 7.8)

We want to know how many measurements in the list are within 1.5 standard deviations of the mean

Well, using the above logic, 0.45 represents 1.5 units of standard deviation.
So, 1.5 units of standard deviation below the mean equals 7.65 (8.1 - 0.45 = 7.65)

Similarly, 1.5 units of standard deviation above the mean equals 8.55 (8.1 + 0.45 = 8.55)
So, any measurement that is between 7.65 ounces and 8.55 will be within 1.5 standard deviations of the mean.

In the given list of measurements, the following meet this requirement:

7.51 8.22 7.86 8.36
8.09 7.83 8.30 8.01
7.73 8.25 7.96 8.53

Correct Answer: E

Approach Solution 2:
So:

M= 8.1
SD= 0.3

1.5 SD= 1 SD plus 1/2 SD, which is 0.3+0.15=0.45 SD.

Lets add and subtract 3SD or 0.45 from the M or 8.1:

8.1 - 0.45 = 7.65
8.1 + 0.45 = 8.55

So, we are looking for numbers in this range 7.65 These are 11 of the 12 numbers

Correct Answer: E

Approach Solution 3:
Mean = 8.1

Standard Deviation = 0.3
within 1.5 standard deviation of the mean

i.e. 1.5 standard deviation ABOVE the mean = 8.1+1.5*0.3 = 8.55
and 1.5 standard deviation BELOW the mean = 8.1-1.5*0.3 = 7.65

Except first value 7.51, all other values are within the range hence, 11 values fall within 1.5 SD from Mean

Correct Answer: E

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