standard deviation Question - Could someone please explain??

[alexkhalaf2000]

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 amount, 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, how many of the 12 listed amounts are within 1.5 standard deviations of the mean?

A- 4
B - 6
C - 9
D- 10
E- 11

[jnelson0612]

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 amount, 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, how many of the 12 listed amounts are within 1.5 standard deviations of the mean?

A- 4
B - 6
C - 9
D- 10
E- 11

Hi Alex, sure.

A standard deviation just shows how spread out members of a population are. A small standard deviation means that most members are close to the mean; a large one means that they are more spread out overall.

An example of standard deviation is seen in IQ scores. The mean IQ score is 100, with a standard deviation of 15. That means that the majority of the population is within one standard deviation on either side: the mean minus the standard deviation and the mean plus the standard deviation. Thus, most members of the set are in the range of 85 to 115. To get the range for two standard deviations from the mean, I simply take 2 * 15 = 30 and then subtract and add that to the mean to get the endpoints. Thus, the range of two standard deviations is 70 to 130.

In this case, the mean is 8.1 and the standard deviation is .3. Thus, 1.5 standard deviations would be 1.5 * .3, or .45. We subtract and add that to the mean to get the range of 1.5 standard deviations. 8.1 - .45 is 7.65; 8.1 + .45 is 8.55. Therefore, 7.65 to 8.55 is the range of 1.5 standard deviations from the mean.

Of the twelve values listed, I only see one (7.51) outside this range. Because of that, the answer is E, 11.

[PFWinkler]

Thank you. Very helpful. I actually MISREAD this problem....

Here we are looking for how many values are WITHIN both 1.5stdv above AND below the mean. I only took values ABOVE the mean and got it WRONG.

This is a critical part of the question, ie: what we are looking for 'how many of the 12 listed amounts ARE WITHIN 1.5 stdv of the mean'

[RonPurewal]

yep.
as always, the fix for that kind of thing is to SLOW DOWN when you are reading the problem statement.

as the US Marines say... slow is smooth, and smooth is fast. NEVER rush through reading the words!
if you try to "hurry" the reading, it will just take you longer anyway (you'll have to go back and re-read / "pick up after yourself" often enough to negate any time benefits you might reap ... and more).