(part 3)

c. Find the 90^th percentile. For each problem or part of a problem, draw a new graph. Draw the x-axis. Shade the area that corresponds to the 90^th percentile. This time, we are looking for a score that corresponds to a given area under the curve.

Let k = the 90^th percentile. The variable k is located on the x-axis. P(x < k) is the area to the left of k. The 90^th percentile k separates the exam scores into those that are the same or lower than k and those that are the same or higher. Ninety percent of the test scores are the same or lower than k, and 10 percent are the same or higher. The variable k is often called a critical value.

We know the mean, standard deviation, and area under the normal curve. We need to find the z-score that corresponds to the area of 0.9 and then substitute it with the mean and standard deviation, into our z-score formula. The z-table shows a z-score of approximately 1.28, for an area under the normal curve to the left of z (larger portion) of approximately 0.9. Thus, we can write the following:

Multiplying each side of the equation by 5 gives

Adding 63 to both sides of the equation gives

Thus, our score, k, is 69.4.

The 90^th percentile is 69.4. This means that 90 percent of the test scores fall at or below 69.4 and 10 percent fall at or above. To get this answer on the calculator, follow this next step: