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Normal distribution (2) When it is not the standard normal distribution The Normal Distribution WRITTEN : X ~ N ( , ) 2 … which means the continuous random variable X is normally distributed with mean and variance 2 (standard deviation ) The Standard Normal Distribution • The random variable is called Z • Z is called the standard normal distribution – its mean is 0 – standard deviation is 1 • The distribution function is denoted by • Area under the curve = probability Z ~ N (0,1) (Z) The Standard Normal Distribution Z ~ N (0,1) The probabilities are given by the area under the curve (-1.6) = P(Z<-1.6) =0.0548 By symmetry: (1.6) =1 - (-1.6) P(Z<-1.6) = 1 - P(Z<1.6) Probability above 75? Probability student scores higher than 75? 0.08 0.07 Density 0.06 0.05 P(X > 75) 0.04 0.03 0.02 0.01 0.00 55 60 65 70 75 Grades 2 X ~ N (70,5 ) 80 85 The Normal Distribution X ~ N ( , ) 2 The Standard Normal Distribution Z ~ N (0,1) •Tables are for standardised Z •May want to find other solutions (given and 2) •The normal distributions must be ‘standardised’ •However, GDCs can handle either Standardising X ~ N ( , ) 2 Use the transformation Z Z ~ N (0,1) X …. then, use probability table for Z Probability above 75? Probability student scores higher than 75? 0.08 0.07 2 X ~ N (70,5 ) Density 0.06 0.05 P(X > 75) 0.04 P(X>75) 0.03 0.02 0.01 0.00 55 60 65 70 Grades Z X 1 - P(X<75) 75 80 85 = 1 - P(X<75) 75 70 Z 5 1 - P(Z<1) = 1 - 0.8413 = 0.1587 Probability between 65 and 70? 2 X ~ N (70,5 ) 0.08 0.07 Density 0.06 0.05 P(65 < X < 70) 0.04 0.03 0.02 0.01 0.00 55 60 65 70 75 80 85 Grades P(65<X<70) = P(X<70) - P(X<65) Probability between 65 and 70? 2 X ~ N (70,5 ) 0.08 0.07 Density 0.06 0.05 P(65 < X < 70) 0.04 0.03 0.02 0.01 0.00 55 60 65 70 75 80 85 Grades P(65<X<70) = P(X<70) - P(X<65) Z X P(-1<Z<0) 70 70 65 70 Z Z 5 5 P(Z<0) - P(Z<-1) P(Z<0) - [1- P(Z<1)] 0.5 - [1 - 0.8413] = 0.3413 Probability between 65 and 70? Why not GDC? 0.08 0.07 Density 0.06 0.05 normalcdf(lower bound, upper bound, mean (), standard deviation ()) P(65 < X < 70) 0.04 0.03 0.02 0.01 0.00 55 60 65 70 75 80 85 Grades 2 X ~ N (70,5 ) P(65<X<70) 2nd distr 2 normalcdf(65, 70, 70, 5) Z ~ N (0,1) P(-1<Z<0) 2nd distr 2 normalcdf(-1, 0) (if you close bracket it assumes ‘Z’) Probability above 75? Probability student scores higher than 75? 0.08 0.07 2 X ~ N (70,5 ) 0.06 Density 0.05 P(X > 75) 0.04 0.03 0.02 0.01 0.00 55 60 65 70 75 80 85 Grades normalcdf(lower bound, upper bound, mean (), standard deviation ()) No upper bound!!!! 2nd distr 2 normalcdf(75, E99, 70, 5) Very very big Probability below 65? 0.08 0.07 2 0.06 X ~ N (70,5 ) Density 0.05 0.04 0.03 0.02 P(X < 65) 0.01 0.00 55 65 75 85 Grades normalcdf(lower bound, upper bound, mean (), standard deviation ()) No lower bound!!!! 2nd distr 2 normalcdf(-E99, 65, 70, 5) Very very small