need help with statistics please answer quick

Question

Scores of an IQ test have a​ bell-shaped distribution with a mean of

100 and a standard deviation of 20.

Use the empirical rule to determine the following.

What percentage of people has an IQ score greater than

160?

Answer 1

Let's first make certain that you have the basics. First, "mean of 100" & "standard deviation of 20" refer directly to IQ scores. The "mean" (average) IQ score is 100, while each standard deviation (SD) represents 20 points on the IQ scale. Next, you must remember the percentage of IQ scores within each standard deviation. In a "normal" distribution (bell-shaped distribution), half of all scores will be above & and half will be below the mean (average) IQ score, a little over 2/3 of all IQ scores will fall between -1 SD and +1 SD, and almost 1/6 of all scores will be +2 SD and higher. This aspect is more like geometry: measure the applicable parts of the curve, and that answer will represent the percentage of people with the particular IQ score. It is useful for you to draw the curve and mark the SDs (-3, -2, -1, +1, +2, +3), then mark the percentages of scores under each part of the curve. I'll show you the answer, but do yourself a favor - study the curve more closely and learn how the distribution is "normalized" or theoretical, so that you can see how the "area" of the curve is standardized for easy use.

Believe it or not, it really is this simple:

1. Scores above 160, are 60 or more above the mean of 100.
2. The standard deviation in this problem uses increments of 20 IQ points. Therefore, IQ scores 80-120 are within the first standard deviation (100-20 through 100+20). Scores 121-140 are within the 2nd SD, and scores 141-160 are within the 3rd SD. IQ scores beyond the 3rd SD above 160.
3. It is that 3rd deviation that we need to focus on, and find the "area" of the space under the curve - but only up to the 3rd SD. What percentage of scores are represented by that portion of the curve? Once we find that percentage, we simply subtract from 100%.
4. Scores above 160 are in far right end of the curve, beyond the +3 SD mark. Now let's add: The left (lower) half of the curve is 50%. Up to +1 SD adds another 34.13%. That's a total of 84.13% with IQ scores up to 120.
5. Now we measure up to +2 SD and add 13.6% to our total. This tells us that 97.73% of all IQ scores are 140 and below.
6. Finally, we add 2.14% to our total, as we measure the rest of the way up to +3 SD, and the total is 99.87%. This means more than 99% of all IQ scores are below 160, so we subtract that percentage from 100%. The answer in this case is 0.13%, but some "normal distribution" charts will vary in the percentages for each portion of the curve. Nonetheless, Your answer here should be within the range of 0.13% up to perhaps 0.28% - depending upon the particular chart you use. Fewer than 0.3% of all IQ scores are above 160.