Johannah I. answered • 03/17/20
Experienced Math Tutor
Hi,
There's a few ways you can go about this problem. 1, you can normalize the data by finding z-scores for 80, 50, and 40. Doing so will allow you to use the empirical rule or a normal distribution table to calculate the area, which can be converted to a percentage. You can also use the normalcdf function on a graphing calculator which allows you to find the area under the curve.
If I normalize 80:
(80-50)/15 = 2
Using the empirical rule, from 2 standard deviations above the mean and below would be 97.5%
Using a normal distribution table (https://www.math.arizona.edu/~rsims/ma464/standardnormaltable.pdf), the area to the left of 2 is .97725 or 97.725%
Using the normalcdf function:
normalcdf(-1E99, 2, 0,1) = .977249938 or 97.7249938%
OR
normalcdf(-1E99,80,50,15) = .977249938 or 97.7249938%
You can use these same methods to evaluate the other questions, with the exception of the final question. The final question asks what is the top 25%? In the prior problems, you were given values and being asked to find percentages. This is just the opposite. You have been given a percentage and are being asked to provide a value.
You can do this using the normal distribution table or the invNorm function on your calculator.
The normal distribution table provides the area to the left of given z-values. Since we're looking for the top 25%, we want the area to the right. So, instead of using 25%, I'm going to use 75%. (1 - .25 = .75). Use the table to find .75 in the middle of the chart. The closest value I can find is 0.74857. This value corresponds with the z-score 0.67. I can use the z-score formula to figure out the score which differentiates the top 25% from the bottom 75%:
.67 = (x-50)/15
10.05 = x-50
60.05 = x
Using the invNorm function:
invNorm(.75,50,15) = 60.1173
(Note: .75 is used again because invNorm also asks for the area to the left).
Hope this helps!
