How to solve s.d. question

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

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 1,000 cups dispensed by the vending machine, the 12 listed amounts, in ounces, were selected from the data. If the 1,000 recorded amounts have a mean of 8.1 ounces and a standard deviation of 0.3 ounce,howmany of the 12 listed amounts are written 1.5 standard deviations of the mean?

four
six
nine
ten
eleven

Could someone please explain the rationale of solving this by using Z-score? For the experts out there, how much stats is on the real GMAT? I was quite taken aback that this question requires knowledge of z-score to solve. The s.d. chapter is pretty simple in OG, but are going to end up using the t-test too?

Answer is 11

Hi OneTwoThreeFour,

Although this has a z-score element, test-takers don't need to know anything about z-scores (knowledge of t-tests is unnecessary too).

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 abovethe 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

So the answer is eleven