1. Normal Distribution To understand the normal distribution To be able to find probabilities given the Z score To be able to find the Z score given the probability

2. Most commonly observed probability distribution 1800s, German mathematician and physicist Karl Gauss used it to analyse astronomical data Sometimes called the Gaussian distribution in science.

3. Normal Distribution Occurs naturally(e.g. height, weight,..) Centres around the mean Often called a bell curve

4. Normal Distribution Spread depends on standard deviation Percentage of distribution included depends on number of standard deviations from the mean

5. Properties of Normal Distribution Symmetrical Area under curve = 1

6. Standard Normal Distribution Mean ( =0 Standard deviation ( )=1

7. Standard Normal Distribution Z-scores are a means of answering the question ``how many standard deviations away from the mean is this observation?'' Tables are provided to help us to calculate the probability for the standard normal distribution, Z

8. Find P(Z<1.25) Tables give us P(Z<z) It is vital that you always sketch a graph P(Z<1.25) = 0.8944

9. Find P(Z>1.25) Tables give us P(Z<z) It is vital that you always sketch a graph P(Z>1.25) = 1- 0.8944 = 0.1056

10. a) Find P(Z < 1.52) It is vital that you always sketch a graph b) Find P(Z > 2.60) c) Find P(Z < -0.75) d) Find P(-1.18 < Z < 1.43)

11. a) Find P(Z < 1.52) SOLUTIONS P(Z < 1.52) = 0.9357

12. SOLUTIONS P(Z > 2.60) = 1 - 0.9053 = 0.0047 b) Find P(Z > 2.60)

13. SOLUTIONS P(Z 0.75) c) Find P(Z < -0.75) P(Z > 0.75) = 1 – P(Z < 0.75) P(Z > 0.75) = 1 – 0.7734 = 0.2266

14. SOLUTIONS P(Z<1.43) = 0.9236 d) Find P(-1.18 < Z < 1.43) P(Z>1.18) = 1-0.881 P(Z>1.18) = 0.119 P(-1.18<Z<1.43) = 0.9236 - 0.119 = 0.8046

15. Reversing the process Given the probability find the value of a in P(Z<a) P(Z<1.25) = 0.8944P(Z<-0.25) = 0.4013 If the probability is >0.5 then a is positive If the probability is <0.5 then a is negative

16. a) P(Z < a) = 0.7611 It is vital that you always sketch a graph b) P(Z > a) = 0.0287 c) P(Z < a) = 0.0170 d)P(Z > a) = 0.01 ASK ABOUT THIS ONE

17. SOLUTIONS a = 0.71 a) P(Z < a) = 0.7611 0.7611

18. SOLUTIONS a = 1.9 b) P(Z > a) = 0.0287 0.02870.9713

19. SOLUTIONS z = 2.12 so a = -2.12 c) P(Z < a) = 0.0170 0.0170 < 0.5 so a is negative 0.01700.9830

20. SOLUTIONS Use percentage points of normal distribution table which gives P(Z>z) d)P(Z > a) = 0.01 a = 2.3263

21. Normal distribution calculator