1. Normal distribution (2) When it is not the standard normal distribution

2. The Normal Distribution WRITTEN : … which means the continuous random variable X is normally distributed with mean  and variance  2 (standard deviation  )

3. 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)

4. The Standard Normal Distribution 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)

5. Probability above 75?

6. The Normal Distribution The Standard Normal Distribution 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

7. Standardising …. then, use probability table for Z Use the transformation

8. Probability above 75? P(X>75) 1 - P(X<75) = 1 - P(X<75) 1 - P(Z<1) = 1 - 0.8413 = 0.1587

9. Probability between 65 and 70? P(65<X<70) = P(X<70) - P(X<65)

10. Probability between 65 and 70? P(65<X<70) = P(X<70) - P(X<65) P(-1<Z<0) P(Z<0) - P(Z<-1) P(Z<0) - [1- P(Z<1)] 0.5 - [ 1 - 0.8413] = 0.3413

11. Probability between 65 and 70? P(65<X<70) P(-1<Z<0) 2nd distr normalcdf(lower bound, upper bound, mean (  ), standard deviation (  )) Why not GDC? normalcdf(65, 70, 70, 5) 2 2nd distr normalcdf(-1, 0) 2 (if you close bracket it assumes ‘Z’)

12. Probability above 75? No upper bound!!!! 2nd distr normalcdf(75, E99, 70, 5) 2 Very very big normalcdf(lower bound, upper bound, mean (  ), standard deviation (  ))

13. Probability below 65? No lower bound!!!! 2nd distr normalcdf(-E99, 65, 70, 5) 2 Very very small normalcdf(lower bound, upper bound, mean (  ), standard deviation (  ))