1. Finding the Normal Mean and Variance © Christine Crisp “Teach A Level Maths” Statistics 1

2. Finding the Normal Mean and Variance "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Statistics 1 AQA Normal Distribution diagrams in this presentation have been drawn using FX Draw ( available from Efofex at www.efofex.com )www.efofex.com

3. Finding the Normal Mean and Variance The standardising formula to change values from a random variable X into Z values is We can also use this formula to find either  or  ( or both ) provided we know, or have enough information to find, the other unknowns.

4. Finding the Normal Mean and Variance Solution: and means where Using the Percentage Points of the Normal Distribution table, So, It doesn’t matter whether we find the z value first or use the standardising formula first. e.g.1 Find the values of  in the following:

5. Finding the Normal Mean and Variance Tip: It’s easy to make a mistake and add instead of subtract or vice versa so check that your answer is reasonable by comparing with the information in the question. We had so x = 50 The mean is clearly less than 50 so the answer is reasonable.

6. Finding the Normal Mean and Variance and means where Using the Percentage Points of the Normal Distribution table, So, e.g.2 Find the values of the unknown in the following: Solution:

7. Finding the Normal Mean and Variance Exercise 1. Find the values of the unknowns in the following: (a) and (b) and

8. Finding the Normal Mean and Variance Solution: (a) and where Using the Percentage Points of the Normal Distribution table, So,

9. Finding the Normal Mean and Variance Solution: where Using the Percentage Points of the Normal Distribution table, So, (b) and

10. Finding the Normal Mean and Variance In the next two examples  and  are both unknown. The 2 nd of these is set as a problem rather than a straightforward question.

11. Finding the Normal Mean and Variance Solution: Using the Percentage Points of the Normal Distribution table, e.g. 3 Find  and  if and and We have and N.B. is negative So, and

12. Finding the Normal Mean and Variance and We must solve simultaneously: and We can change the signs in the 1 st equation and add:Adding: Substitute into either equation to find  : e.g.

13. Finding the Normal Mean and Variance Solution: Let X be the random variable “length of rod (cm)” e.g.4 The lengths of a batch of rods follows a Normal distribution. 10% of the rods are longer than 101 cm. and 5% are shorter than 95 cm. Find the mean length and standard deviation. and So, 050)( 1  zZP

14. Finding the Normal Mean and Variance and Solving simultaneously: We can change the signs in the 2 nd equation and add: Substitute into either equation to find  : Adding: e.g.4 The lengths of a batch of rods follows a Normal distribution. 10% of the rods are longer than 101 cm. and 5% are shorter than 95 cm. Find the mean length and standard deviation.

15. Finding the Normal Mean and Variance SUMMARY  To find one parameter ( mean or standard deviation ) of a variable with a Normal Distribution we need to know the other parameter and one “pair” of values of the variable and corresponding percentage or probability.  To find both the mean and standard deviation we need to know two “pairs” of values of the variable and corresponding percentages or probabilities.

16. Finding the Normal Mean and Variance Exercise and 1.Find the values of  and  if 2. A large sample of light bulbs from a factory were tested and found to have a life-time which followed a Normal distribution.  of the bulbs failed in less than  hours and  lasted more than  hours. Find the mean and standard deviation of the distribution.

17. Finding the Normal Mean and Variance Solution: Using the Percentage Points of the Normal Distribution table, We have and So, and 1. and and

18. Finding the Normal Mean and Variance Solving simultaneously: and We can change the signs in the 1 st equation and add: Adding: Substitute into either equation to find  : e.g. and

19. Finding the Normal Mean and Variance Solution: Let X be the random variable “life of bulb (hrs)” So, 2. A large sample of light bulbs from a factory were tested and found to have a life-time which followed a Normal distribution.  of the bulbs failed in less than  hours and  lasted more than  hours. Find the mean and standard deviation of the distribution. We know that and So, and

20. Finding the Normal Mean and Variance Using the Percentage Points of the Normal Distribution table, So, and z 1 is negative Adding:

22. Finding the Normal Mean and Variance The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

23. Finding the Normal Mean and Variance The standardising formula to change values from a random variable X into Z values is We can also use this formula to find either  or  ( or both ) provided we know, or have enough information to find, the other unknowns.

24. Finding the Normal Mean and Variance Solution: (a) and means where Using the Percentage Points of the Normal Distribution table, So, It doesn’t matter whether we find the z value first or use the standardising formula first. e.g.1 Find the values of the unknowns in the following:

25. Finding the Normal Mean and Variance Tip: It’s easy to make a mistake and add instead of subtract or vice versa so check that your answer is reasonable by comparing with the information in the question. We had so x = 50 The mean is clearly less than 50 so the answer is reasonable.

26. Finding the Normal Mean and Variance and means Using the Percentage Points of the Normal Distribution table, So, e.g.2 Find the values of the unknown in the following: Solution: where

27. Finding the Normal Mean and Variance Solution: Let X be the random variable “length of rod (cm)” e.g.3 The lengths of a batch of rods follows a Normal distribution. 10% of the rods are longer than 101 cm. and 5% are shorter than 95 cm. Find the mean length and standard deviation. and So, 050)( 1  zZP

28. Finding the Normal Mean and Variance Solving simultaneously: We can change the signs in the 2 nd equation and add: Substitute into either equation to find  : Adding: