Continuous Random Variables /Normal Module VIF 1 Continuous Random Variables and Normal Variables Module VIF Grenith J. Zimmerman, Ph.D. Associate Dean for Research Copyright 2012 School of Allied Health Professions 2 Continuous Random Variables Can take on uncountably infinitely many values (an interval) on the real line Probabilities are determined by areas under a density function curve. 3 Density Curves Always on or above the horizontal axis Have area exactly 1 underneath curve Area under the curve and above any range of values is the proportion of all observations that fall in that range a b X 1 Continuous Random Variables /Normal Module VIF 4 Density Curves The median of a density curve is the equal-areas point, the point that divides the area under the curve in half. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. 5 The Normal Distribution standard deviation mean 6 The Normal Distribution Knowing the mean () and standard deviation () allows us to make various conclusions about Normal distributions. 2 Continuous Random Variables /Normal Module VIF 7 Three normal distributions. Curves A and B have the same standard deviation, but different means; curves B and C have the same means, but the standard deviation of B is larger than that of C 8 689599.7 Rule for Any Normal Curve 95% 68% 99.7% 9 Time for a stroke patient to complete activities of daily living =8.5 min = 2 min. 68% 6.5 8.5 95% 10.5 8.5 4.5 12.5 99.7% 2.5 8.5 14.5 3 Continuous Random Variables /Normal Module VIF 10 Family of Normal Distributions Two Parameters: , : Any value : Any positive value 11 Standard Normal Distribution A normal probability distribution that has a mean of 0 and a standard deviation of 1. 4 ** see power point and power point notes, 2 attachments. This will provide info on the ?'s. Continuous Random Variables/Normal Variables and Calculating Z Probabilites 1) Which of the following statements are true for the normal distribution? 1. The area under the normal curve is 1. 2. The location of the normal curve is determined by its mean. 3. The shape of the normal curve is determined by its variance 4. Probabilities for the normal curve are calculated by finding the area under the curve above the interval of values. 1 and 2 are true 1,2 and 4 are true 1, 2, and 3 are true 2) The standard normal distribution has mean equal to . 3) Match the probabilities to the probability statements. P[Z < 2.52] = [ Choose ] P[Z > 1.34] = [ Choose ] and standard deviation equal to P[ 1.60 < Z < 2.85] = [ Choose ] 4) If the P[ Z< ?] = .7734 Z is _______ 5) Find ? if the P[Z > ?] = .9394. The Zvalue for ? is ______