NORMAL PROBABILITY DISTRIBUTIONS The Most Important Probability Distribution in Statistics Total area =1; symmetric around µ

The effects of m and s X 3 6 9 12  µ = 3 and  = 1 How does the standard deviation affect the shape of f(x)? The effects of m and s s= 2 s =3 s =4 How does the expected value affect the location of f(x)? m = 10 m = 11 m = 12 X 8 3 6 9 12  µ = 3 and  = 1

µ = 3 and  = 1  X 3 6 9 12 µ = 6 and  = 1  X 3 6 9 12

X 8 3 6 9 12  µ = 6 and  = 2 X 8 3 6 9 12  µ = 6 and  = 1

P(6 < X < 8) µ = 6 and  = 2 X 3 6 9 12 X Probability = area under the density curve P(6 < X < 8 ) = area under the density curve between 6 and 8. 6 8

Standardizing  

Standard Normal Distribution Z 1 2 3 -1 -2 -3 .5 .5 .5 0.452

P(z  -1.85) -1.85 A1 A2 z -1.18 2.73 P(-1.18  z  2.73)