mean = 63 ounces
SD = 7 ounces
a) 68% of the widget weights lie between 56 and 70
Mean +/- SD
1 SD = 68% of values = 63 +/- 7
2 SD = 95% of values = 63 +/- 14
3 SD = 99.7% of values = 63 +/- 21
This SD-to-Percent conversion includes all values within a set SD Range.
b) What percentage of the widget weights lie between 49 and 70 ounces? 81.8%
63-49 = 14 70-63 = 7
= 2 SD Left = 1 SD Right
= 34.1% + 13.6% = 34.1%
= 47.7%
Total = 47.7% + 34.1%
= 81.8%
This uses the previous SD-to-Percent conversion for 2 numbers. The SDs for these are directional, thus 49 includes values within 2 SD of the mean on the left of the chart only. 70 encompasses values 1 SD to the right of the mean. Once we know what the value of each subsection is, adding them together yields the final value.
c) What percentage of the widget weights lie above 42 ?
This calculation is the exact opposite of the other two. Where they were concerned with the values lying within the SD range, this problem is looking for everything outside of that area. The two main points to check are whether your result should come from inside or outside of the box (the curve) and whether it should be cut in half. All parts of the chart add up to 100%, so by subtracting any SD from 100, you can find the "outside the box" value. If the problem specifies it as either above or below (but not both above and below) a value, then the result should be divided in half.
63 - 42 = 21
22 / 7 = 3 SD
3SD = 99.7% (Inside the box)
100 - 99.7 = 0.3% (outside the box)
0.3 / 2 = 0.15% (only one side)
