1. Top, Middle, & Bottom Cutoff Scores……

2. OBJECTIVE
Find actual values using a normal distribution.

3. RELEVANCE
Find probabilities and values of populations whose data can be represented with a normal distribution.

4. Sometimes you will need to find an actual score when you are given a certain area under the curve.
To do this, you’ll find your z-score and then substitute it back into the z-score formula to find the value (“x”).

5. Example……
An exclusive college will only take the top 10% of applicants based on an entrance test. If the test has a mean of 500 and a st. deviation of 100, find the cutoff score a student would need to make on the test to be admitted to the college.

6. 1st find the z based on the area of “top 10%.”
InvNorm( ) =
InvNorm(0.90) =
1.28.
Next, substitute that value into the z-score formula to figure the x using algebra and proportions.

7. You Try
Suppose that replacement times for washing machines are normally distributed with a mean of 8.4 years and a standard deviation of 2 years. Find the replacement time that separates the top 18% times.

8. 0.1800
8.4
z to x

9. Example……
Researchers want to select people in the middle 60% of the population based on their blood pressure. If the mean blood pressure is 120 and the st. deviation is 8, find the upper and lower readings that would qualify.

10. 1st find the z-score based on “middle 60%.”
0.6000/2 =
InvNorm(0.5 – 0.3)
z = + and – 0.84.
Next, substitute both z values into the z-score formula.

12. You Try
The weights of certain machine components are normally distributed with a mean of 8.01 g and a standard deviation of 0.06 g. Find the two weights that are the boundaries for the middle 90%.

13. 0.9000
0.05
0.05
8.01

14. Example…….
A teacher wants to start a reading class for the bottom 30% of students who take a reading test. The average score on the test is 400 and the st. deviation is 5. Find the cutoff score.

15. Find the z-score first for the “bottom 30%.”
InvNorm(0.3000)
Z = -0.52
Plug in to z-score formula.

16. You Try
Scores on an English test are normally distributed with a mean of 37.3 and a standard deviation of 8. Find the score that separates the bottom 41%.

17. 0.4100
-z to x
37.3