Tim D. answered • 05/24/21
Because the scores are normally distributed we can start by drawing a normal curve, cutting it down the middle and labelling the point along the horizontal scale with the value of the mean, in this case 500. Next, noticing that the standard deviation is 90, we can subtract 90 from 500 to get 410, which is the value to the left of 500 by one standard deviation (s.d.). Since the borderline value we are wanting is 400, that is slightly below 410, so make a mark slightly to the left of 410, and label it 400, and cut the curve vertically at 400.
We want to find the area to the left of that cut. We can find the Z-score of 400, using Z = (400 - mean)/s.d. = (400 - 500)/90 = -100/90 = -1.11 rounded to two places. The Z-score tells us that 400 is 1.11 standard deviations below 500. The Z-score is used to read a Standard Normal table, or Z-table, which gives 0.1333 as the answer.
Whenever the question asks for area below a value, these steps are enough. But if the question asks for scores above a given value, then we need to remember that the decimal from the Z-table is area to the LEFT of the value, and above implies area to the RIGHT of that value, so we must take 1 minus the decimal to get our answer in that case. (Since total area under the curve is 1)