.8385
To answer this, all you really need is a few definitions (and a graphical understanding will help!). The 68-95-99.7 rule is basically one that says, within a normally distributed set of data:
- 68% of the scores/observations lie within 1 standard deviation of the mean
- 95% of the scores lie within 2 standard deviations of the mean
- 99.7% lie within 3 standard deviations of the mean
In general, that idea looks something like this: mean ± k・(standard deviation). Google 68-95-99.7 and you'll get thousands of normal curves showing you what this looks like visually.
Specifically your data has a mean of 100 and a standard deviation of 18. Just looking at what we already went over, that means 68% of your scores are between 100 ± 1・18, so 68% of the scores are between 82 and 118. Now, what your problem is asking for is the scores between 46 and 118; we found 118 using the 68-95-99.7 rule, but maybe we have to go another more standard deviation out to run into 46.
100 - 18 = 82 (not low enough)
82 - 18 = 64 (note that we're now 2 standard deviations away from the mean, but we still haven't found 46)
64 - 18 = 46 (here it is! 3 standard deviations away)
So what we found out is that 46, the lower bound the question asks for, is 3 standard deviations away from the mean, and that the upper bound, 118, is only one standard deviation away. All you have to do now is get crafty with simple operations. Once again, drawing this out might help you more! A lot of these problems are much easier when visualized, but regardless:
Imagine the mean is your home base. To get to 118, you go up 1 standard deviation. If 68% of observations are found within 1 standard deviation (mean ± standard deviation), what we need to do to ONLY find from the mean up to 118 is divide 68% in half, and see that it's 34%. This gives us just the right section of the curve that we need, and now all we have to worry about is the percentage on the left of the mean.
Well, we figured that 46 is 3 standard deviations out, but remember that we're only considering the left half of the normal curve now, so what we'd do in this situation is say to ourselves... "I know that the score I want to include is three standard deviations out, so that's related to 99.7%, but I'm only worried about the left half of this, and 99.7% considers the whole curve. I'll cut that in half, and see it's 49.85%. Cool."
So now what we have is that, between your two scores, 46 and 118, to the left of the mean we're looking at 49.85%, and to the right were looking at 34%. To get the total amount between those two, all you have to do is add! Your question calls for decimal places, so...
.34 + .4985 = .8385.
Before you barge onwards with that answer, pause and make sure that's logical. For a range as wide as 46 to 118, holding 83.85% of the observations sounds pretty alright.
