4B: Normal probability distributions 2/23/07

Review Questions

  1. Normal random variables have two parameters. Name these.
  2. How many different Normal curves are there?
  3. The area under the Normal curve sums to exactly ___ . 
  4. What percentage of the area under a Normal curve falls in the range µ ± s? µ ± 2s? µ ± 3s?
  5. Pr(Z < -2)   _____ . [Use 68-95-99.7 rule; does not require use of a Normal table.]
  6. For a continuous random variable, why is Pr(X < x) = Pr(X x)? [Answer requires understanding of limits.]
  7. What is the 25th percentile on a standard Normal distribution? [Answer requires use of Standard Normal table.]
  8. How do you identify points of inflection on a Standard Normal curve?

Exercises

4B.1 Standard Normal probabilities. Use your Z table to find the probabilities requested below. Draw the Standard Normal Z distribution in each instance. Practice making bell-shaped curves with center of 0. (The mean of all Z distributions is 0.) Curves should be scaled so they have points of inflection one standard deviation above and below the mean.  Curves should also form asymptotes as they gets further from the center of their distribution. Shade the appropriate area under the curve for each problem. Accurate sketches will give you a better sense of how the Standard Normal curve is used to model probabilities.

(A) Pr(Z < - 0.64)
(B) Pr(Z > -0.64)
(C) Pr(Z < 1.65) 
(D) Pr(-0.64 < Z < 1.65)

4B.2 More standard Normal probabilities. Find the percent of the Standard Normal (Z) distribution that is:

(A) below -1.42
(B) above 1.42
(C) below 1.25
(D) between -1.42 and 1.25

4B.3 Hospice stay. Lengths of stay in a particular hospice is approximately Normally distribution with mean 14 days and standard deviation 3 days.

(A) What proportion of stays will be less than or equal to 10 days? 
(B) What proportion will have a length of stay that is greater than 10 days?
(C) What proportion will have a length of stay greater than 18 days?
(D) What proportion will have a stay between 18 and 10 days? 

4B.4 Death penalty case. An inmate on death row in the state of Illinois has an IQ of 51. What percent of people have an IQ of this value or less?

4B.5. Heights of 11--year old boys.  The height of 11-year-old boys varies according to a Normal distribution with mean 146 centimeters with standard deviation 8 centimeters. 

(A) Draw a Normal curve that corresponds to this distribution. Locate the center of the distribution on the horizontal axis and write the value of the mean at this location.
(B) Locate the points of inflection on the curve. This is where the slopes of the curve changes. Mark the horizontal below each inflection point. These points are 1 standard deviation above and below the mean. Write these values by the tick marks. What percentage of heights lie between thee values? 
(C) Use the 68-95-99.7 rule to determine the range of heights that capture the middle 95% of values for this random variable. Mark these values on the Normal curve. .
(D) How tall are the tallest 2.5% of 11-year old boys?

4B.6 Alzheimer brains. Dusheiko (1973) studied the pathological anatomy of Alzheimer's disease. In his study he found that the weight of brains of cases varied according to a Normal distribution with mean of 1076.80 grams and standard deviation 105.76 grams (Daniel, 1999, p. 114). 

(A) What proportion of Alzheimer cases will have brains that weigh less than 900 grams?
(B) What proportion will have brains that weigh more than 1200 grams?

4B.7 Job satisfaction in nurses. A study of job satisfaction in nurses revealed a mean satisfaction score of 50 and a standard deviation of 10. Assume values vary according to a Normal distribution.

(A) What is the probability that a nurse selected at random has a satisfaction score that is less than 40?
(B) What is the probability a nurse selected at random has a satisfaction score greater than 70?

4B.8 Coliform levels. Coliform levels in water samples taken a stream vary demonstrate a mean of 10 organisms per liter with a standard deviation of 2 organisms. Assume the distribution in samples is approximately Normal. What percentage of sample will contain more than 15 organisms?  

4B.9 Z percentiles. Recall that zp refers to a Standard Normal random variate with a cumulative probability ( left tail) of  p. The z score zp is called pth percentile on the Standard Normal curve. Find the following z percentiles.

(A) z.05 =
(B) z.45 =
(C) z.64 =
(D) z.80 =
(E) z.99 =

4B.10 More Z percentiles. Find the following z percentiles. 

(A) z.10 = 
(B) z.35 =
(C) z.74 =
(D) z.85 =
(E) z.999 =

4B.11 Heart rate. The distribution of resting heart rates in a healthy population is approximately Normal with mean 70 beats per minute and standard deviation 10.What proportion of the healthy population will have a resting heart rate greater than 80 beat per minute?

Key to Odd Numbered Problems                             Key to Even Numbered Problems (may not be posted)