ANSWERS FOR Lab 3, Part II 1. Assume the variable y (age) is normally distributed with a mean of 45 and a standard deviation of 15. a. What proportion of the cases are: i. Younger than 50? ii. Older than 58? iii. Between the ages of 43 and 51? Z-scores for the distribution of individual cases, Mean = 45; standard deviation = 15 Use the formula for Z scores: Z y Y sy a. Determining proportions: i. z = 0.33 ; proportion > .33 = .3707; proportion < 50 = (1-.3707) = .6293 The proportion of cases below the age of 50 is .6293. (In percent, 62.93% are below the age of 50). ii. z = 0.87 ; proportion > 0.87 = .1922; proportion > 58 = .1922 The proportion of cases older than 58 is .1922 (19.22% are older than 58 years of age). iii. z = -0.13, 0.40 ; proportion <-0.13 = .4483; proportion > 0.40=.3446; proportion between 43 and 51 = 1 - (.4483 + .3446) = .2071 The proportion of cases between 43 and 51 is .2071 (20.71% are between 43 and 51). b. Find the age (or ages) of the following: HINT: First, look up the proportion in the Z-table to get the z-score, and then solve the equation above for “y” instead of Z. i. which divides the distribution in half. Proportion > (Z=0) = .5. y = 45 Forty-five years (45) is the age which divides the distribution in half (this is a give away question, as the mean is always what divides the distribution in half, on a normal curve). ii. above which 25% of the cases fall. Proportion > (Z=.67) = .25. y= 55.05 years. Twenty-five percent of the cases are above 55.05 years of age. iii. below which 5% of the cases fall. Proportion < (Z= -1.64) = .05. y = 20.4 The age below which 5% of the cases fall is 20.4 years. iv. above which 2.5% of the cases fall. Proportion > (Z=1.96) = .025. Two and a half percent of the cases are older than 74.4 years. v. below which 1% of the cases fall. Proportion < (Z= -2.33) = .01. One percent of the cases are younger than 10.05 years. vi. above which 0.5% of the cases fall. Proportion > (Z=2.58) = .005. One half of one percent (0.5%) of the cases are older than 83.7 years.