1. A distribution of test scores follows the normal distribution with x̅ = 80, s = 12, n= 600. How many scores would you expect to find below a score of 90?

Answer 1

Step1. Conceptualize the question.

Imagine a normal distribution, with a mean of 80 in the middle of that distribution. The score of 90, being above 80, will be to the right of that distribution. Recall that in a normal distribution, the area under the curve for any particular point is the total percentage of data you will find. Therefore, if we would like to know how many scores will be below a score of 90, then we are looking for the area under the curve (starting at a score of 90) which will give us the percentage of data below that score. We then multiply that percentage by the sample size to find the total amount of scores before that point.

Step2. Note your Givens

1. Xbar=80 (Sample Mean
2. S=12 (Sample STD)
3. n=600

Step3: Determine your Z test value (to get this, we take our observation of interest, score of 90, and convert it into a Z test value.

Using the formula

Ztest= (X - XBar) ÷ S/√n

We get

(90 - 80) ÷ 12/√600 = 20.4 = Ztest value

Step 4. Statistical Appendix to derive area under curve fo Z=20.4

P(X<20.4) ≈ 1 (Statisical appendix tables typically go up to Z=3.09, by which the area under the curve is .9990)

Since the area under the curve is virtually 1, then the probability of seeing a score under 90 is approximately 100%.

Scores under 90 = n*Probablity = 600 * 1.00 ≈ 600.