Final exam scores in a mathematics course are normally distributed with a mean of 77 and a standard deviation of 7

Question

Exam score	Z-score	Percentile
81.69
91
	1
		25.14

Based on the above information and a Z-table, fill in the blanks in the table below.

Precision and other notes: (1) Percentiles should be recorded in percentage form to two decimal places.

(2) Note that this problem does not use the rough values of the 68-95-99.7 rule (that is, the empirical rule); instead you must use more precise Z-table values for percentiles.

Answer 1

All of these questions use the z-score formula (x - μ) / σ, which is then keyed to a value on the standard normal table. The first two are a straightforward application of the formula, and they are solved as follows.

z = (81.69 - 77) / 7 = 0.67

On the normal table, the cumulative area to the left of 0.67 is 0.74857, which means that this score is at the 74.857 percentile.

The next z score is z = (91 - 77) / 7 = 2

This cumulative area is 0.97725, or the 97.725 percentile.

The 3rd question gives the z score, so its percentile is easy to find. The area to the left of z = 1 is 0.84134, or a percentile of 84.134. The score in point units is obtained by solving for x in the z-score formula z = (x - μ) / σ. So we plug in the known values and solve for the unknown x:

1 = (x - 77) / 7

7 = x - 77

84 = x

The last one gives us only the percentile. This one is a little tricky because it is below 50%. If your normal distribution table includes negative z scores, it is easy. If not, then you will need to make use of the symmetry of the normal curve.

The z-score associated with the 25.14 percentile (0.2514) is -0.67. Now we can use the same formula as we did with the last calculation and solve for the original exam score:]

-0.67 = (x - 77) / 7

-4.69 = x - 77

72.31 = x.