# Normal Distributions

## Worksheet 3

Open the interactive version of the normal tables. (Opens a separate browser window.)

Obtain some pictures of normal curves. Print them out and use them to help solve problems.

## Exercises

### Draw pictures!

1. The angiogram is a standard diagnostic test used in clinical medicine to detect stroke in patients. This test has some risks for the patient, and several noninvasive techniques have been developed that are hoped to be as effective as the angiogram. One such method utilizes that measurement of cerebral blood flow (CBF) in the brain, since stroke patients tend to have lower levels of CBF than normal. Among healthy people, CBF is normally distributed with mean 75 and standard devitation 17. Patients are classified as being at risk for stroke if their CBF is below 40. What proportion of normal patients will be mistakenly classified as being at risk for stroke?

2. Maple tree diameters in a forest area are normally distributed with mean 10 inches and standard deviation 2.2 inches. Find the proportion of trees having a diameter greater than 15 inches.

3. Our subjects are 35-44-year-old males whose blood pressures are normally distributed with mean 80 and standard deviation 12; N(80, 12). A borderline hypertensive is defined as a person whose diastolic blood pressure is between 90 and 95 mm Hg inclusive; what proportion of subjects are borderline hypertensive? A hypertensive is a person whose diastolic blood pressure is above 95 mm Hg; what proportion of subjects are hypertensive?

4. White blood cell (WBC) count per cubic millimeter of whole blood has approximately the N(7500, 1750) distribution. The lowest 2% of all WBC counts are defined to be probable risks. How low must one's WBC count be to fall in the at-risk group?

5. Glaucoma is a disease of the eye that is manifested by high intraocular pressure. The distribution of intraocular pressure in the general population is approximately normal with mean 16 mm Hg and standard deviation 3.2 mm Hg. If the normal range for intraocular pressure is between 10 and 22 mm Hg, than what proportion of the general population would fall within this range?

6. The resting heart rate for healthy adult horses averages 46 beats per minute with a standard deviation of 8 beats per minute. A horse whose resting heart rate is in the upper 10% of the distribution of heart rates may have a secondary infection or illness that needs to be treated. How fast must a healthy horse's heart be beating to fall into this at-risk group?

## Solutions

### Pictures omitted!

1. Find the area left of Z = -2.06. The answer is: 0.0197.

2. Find the area right of Z = 2.27. The answer is: 0.0116.

3. Find the area between Z = 0.83 and Z = 1.25. The answer is 0.8944 - 0.7967 = 0.0977.

4. The Z value with 2% = 0.02 area left of it is (approximately) -2.05. So, we need the WBC count that is 2.05 times the standard deviation below (because the Z-score is negative) the mean. The answer is 7500 - 2.05 * 1750 = 3913.

5. Find the area between Z = 1.875 and Z = -1.875. The answer is 0.9392 (if you're close--that's fine).

6. The Z value with 0.10 area right of it (and 0.90 area left of it) is 1.28. The answer is 46 + 1.28 * 8 = 56.24 beats per minute.