4B: Normal probability distributions (Key Odd) 

Review Questions

  1. µ (mean) and (standard deviation)
  2. Infinite. (You have a different Normal curve every time you change µ and/or )
  3. 1
  4. 68%, 95%, and 99.5%, respectively.
  5. 0.025 [based on the 68-95-99.7 rule]
  6. Because Pr(X = x) = 0. 
  7.   z.25 = -0.67 [from Z table]
  8.  It is where the curve changes slope, i.e., begins to flatten out, in both directions.

Exercises

4B.1 Standard Normal probabilities 

(A) Pr(Z < - 0.64) = 0.2611 Drawing not shown on the web due to technical difficulties. Students should draw curves, since this helps them make the transition from probabilities to AUCs.
(B) Pr(Z > -0.64) = 1 - 0.2611 = 0.7389
(C) Pr(Z < 1.65) = 0.9505 
(D) Pr(-0.64 < Z < 1.65) = 0.9505 - 0.2611 = 0.6894 [Draw curve, mark z percentiles, shade area between percentiles, look up cumulative probabilities, subtract smaller cumulative probability from larger cumulative probability.] 

4B.3 Hospice stay: X~N(14, 3)

(A) What proportion of stays will be less than or equal to 10 days? 
Standardize the value: z = (10 - 14) / 3 = -1.33. 
Draw curve with landmarks and shading.
Look up the cumulative probability in Z table: Pr(Z <= -1.33) = 0.0918 or about 9.2%. 

(B) What proportion will have a length of stay that is greater than 10 days? 
From the curve you drew in Part A, you can see that this is the complement (right-tail region) of the cumulative probability you just calculated. 
Therefore, subtract the prior probability from 1 to get: 1 - 0.0918 = 0.9082 or about 90.8%.

(C) What proportion will have a length of stay greater than 18 days? 
Standardize the value: z = (18 - 14) / 3 = 1.33. 
Draw the curve with landmarks and shading.
Look up cumulative probability of z in Z table, but notice you need the AUC in the right-tail (not left tail).
Since Pr(Z <= 1.33) = 0.9082, then Pr(Z > 1.33) = 1 - 0.9082 = 0.0918 or about 9.2%.

(D) What proportion will have a stay between 18 and 10 days?
Recall that a value of 18 from this distribution has a z score of 1.33. A value of 10 has a z score of -1.33. 
Draw the curve with these landmarks and shade the area between them.
We've already looked up these cumulative probabilities (0.9082 and 0.0918, respectively). Now subtract the smaller from the larger to find the AUC between these points: 
Pr(10 X 18) = Pr(X < 18) - Pr(X < 10) = 0.9082 - 0.0918 = 0.8164

4B.5 Heights of 11--year old boys

(A) Draw a Normal curve that corresponds to this distribution. Locate the center of the distribution...
 The sketch of the curve is not shown because of technical limitations. Make certain your sketch is accurate (symmetrical, good inflection points, asymptotes) with the center labeled 146 cm..

(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. Points of inflection are located at 146 - 8 = 138 and 146 + 8 = 154. What percentage of heights lie between thee values? 68% of the values like between these points.

(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.
The middle 95% of values is captured by 146 ± (2)(8) = 146 ± 16 = (130, 162).

(D) How tall are the tallest 2.5% of 11-year old boys?
162 centimeters, which is equal to approximately 64 inches (5 foot, 4 inches)

4B.7 Job satisfaction in nurses

(A) z = 40 - 50 / 10 = -1; Pr(Z < -1) = 0.1587 
(B) z = 70 - 50 / 10 =  2; Pr(Z > 2) = 0.0228

4B.9 Z percentiles

(A) z.05 = -1.645 (between -1.64 and -1.65)
(B) z.45 = -0.13
(C) z.64 = 0.36
(D) z.80 = 0.84
(E) z.99 = 2.33

4B.11 Heart rate. X~N(70, 10). What proportion of the population will have a resting heart rate that is 80 or greater? 

z = (80 - 70) / 10 = 1 
Draw curve. Notice that you are looking for the AUC in the right-tail.
Pr(Z 1) = 1 - Pr(Z 1) = 1 - 0.841345 = 0.1587.