NORMAL RANDOM VARIABLES (Normal or gaussian distribution)

Transcription

1 NORMAL RANDOM VARIABLES (Normal or gaussian distribution) Many variables, as pregnancy lengths, foot sizes etc.. exhibit a normal distribution. The shape of the distribution is a symmetric bell shape. The symmetry indicates that the variable is just as likely to take a certain distance below its mean as it is to take the same distance above its mean The shape is determined by the mean µ, the spread by the standard deviation σ. Since the total area must be 1, if the distribution spread gets larger, the height is lower.

2 THE STANDARD DEVIATION RULE AREA=0.68=68% P(µ σ < X < µ +σ) = 0.68 = 68%

3 THE STANDARD DEVIATION RULE AREA=0.95=95% P(µ 2σ < X < µ + 2σ) = 0.95 = 95%

4 THE STANDARD DEVIATION RULE AREA=0.997=99.7% P(µ 3σ < X < µ + 3σ) = = 99.7%

5 TOTAL AREA= =

6 TOTAL AREA= =

7 TOTAL AREA= =

8 Example: Suppose that foot length of adult males is a normal random variable with µ=11 and σ=1.5. a) What is the probability that a randomly chosen adult male will have a foot length between 8 and 14 inches? b) An adult male is almost guaranted (0.997 probability) to have a foot length between what two values? c) The probability is only 2.5% that an adult male will have a foot length greater than how many inches? a) 95% b) 6.5 and 15.5 inches c) 14 inches d) 50% d) What is the probability that a male s foot is shorter than 11 inches?

9 EXAMPLE: Length (in days) of human pregnancy is a normal random variable X with mean 266 and standard deviation 16. a) The probability is 0.95 that a pregnancy will last between:? b) The shortest 16% of pregnancies last less than:? c) The probability of a pregnancy lasting longer than 314 days is:? d) There is a probability of 50% that a pregnancy will last longer than:? a) P(µ 2σ < X < µ+2σ)= b) P(X < µ σ)= c) P(X > 314)=P(X >µ+3σ)= = d) P(X > µ)=50% 266

10 STANDARDIZING VALUES In order to calculate probabilities of a normal random variable, one has to determine how many standard deviations (s.d.) σ below or above the mean a value is. Example: How many s.d. Below or above the mean male foot length is 13 inches? The mean is µ=11, 13 inches is 2 inches above the mean. The s.d. is σ=1.5: It is: = standard deviations above the mean z-score of the random variable (13 inches 11 inches) = inches

11 We say that we have standardized the value of 13. z-score = value µ = x µ σ σ Since σ it is always positive: x above the mean -> z is positive x is below the mean -> z is negative. EXAMPLE: What is the standardized value for a foot length of 8.5 inches? z = = 1.67 It is 1.67 standard deviations below the mean EXAMPLE: a man s standardized foot length is What is his actual foot length in inches? z = x = +2.5 x =11+ ( ) =14.75 inches

12 Z-scores also allow us to compare values of different normal random variables: In general, women s foot length is shorter than men s. Assume that women s foot length follows a normal distribution with µ=9.5 inches and σ=1.2. Paul has a foot length of inches Mary has a foot length of 11.6 inches Which of the two has a longer foot relative to his or her gender group? Paul z = = Mary z = = Mary has a longer foot length relative to her gender distribution.

13 FINDING PROBABILITIES WITH THE NORMAL TABLE Since the normal distribution is symmetric about the mean, the curve of the z-score must be symmetric about 0, and the areas on either side of z=0 are both Z=0 0.5 Following the s.d. Rule, most of the curve falls between z=-3 and z=+3. The normal table lists the number of s.d. a n o r m a l variable z is below or above its mean.

14 z

15 Normal Table 07/03/12 17:28 Ex: What is the probability of a normal random variable taking a value less than 2.8 s.d above its mean (P(Z<2.8))? z Ex: What is the probability of a normal random variable taking a value less than 1.47 s.d below its mean (P(Z<-1.47))? z

16 Ex: What is the probability of a normal random variable taking a value more than 0.75 s.d above its mean (P(Z > 0.75))? Method 1 z P(Z > 0.75) =1 P(Z < 0.75) = =

17 Method 2 By symmetry: P(Z > 0.75) = P(Z < -0.75) = z

18 Ex: What is the probability of a normal random variable taking a value between 1 s.d below and 1 s.d. Above its mean P(-1 < Z < 1)? P( - 1 < Z < 1)=P(Z < 1)- P(Z < - 1)= =