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

( available from Efofex at www.efofex.com ) Statistics 1 EDEXCEL Normal Distribution diagrams in this presentation have been drawn using FX Draw ( available from Efofex at www.efofex.com ) "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"

The standardising formula to change values from a random variable X into Z values is We can also use this formula to find either m or s ( or both ) provided we know, or have enough information to find, the other unknowns.

e.g.1 Find the values of m in the following: and Solution: 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.

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.

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

Exercise 1. Find the values of the unknowns in the following: (a) and (b) and

Solution: (a) and where Using the Percentage Points of the Normal Distribution table, So,

Solution: (b) and where Using the Percentage Points of the Normal Distribution table, So,

In the next two examples m and s are both unknown. The 2nd of these is set as a problem rather than a straightforward question.

e.g. 3 Find m and s if and and Solution: We have and N.B. is negative Using the Percentage Points of the Normal Distribution table, So, and

and and We must solve simultaneously: We can change the signs in the 1st equation and add: Adding: Substitute into either equation to find m: e.g.

e. g. 4 The lengths of a batch of rods follows a Normal distribution 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. Solution: Let X be the random variable “length of rod (cm)” and So, 05 ) ( 1 × = < z Z P So,

e. g. 4 The lengths of a batch of rods follows a Normal distribution 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 Solving simultaneously: We can change the signs in the 2nd equation and add: Adding: Substitute into either equation to find m:

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.

Exercise 1. Find the values of m and s if and and 2. A large sample of light bulbs from a factory were tested and found to have a life-time which followed a Normal distribution. 25% of the bulbs failed in less than 2000 hours and 15% lasted more than 2200 hours. Find the mean and standard deviation of the distribution.

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

and and Solving simultaneously: We can change the signs in the 1st equation and add: Adding: Substitute into either equation to find m: e.g.

2. A large sample of light bulbs from a factory were tested and found to have a life-time which followed a Normal distribution. 20% of the bulbs failed in less than 2000 hours and 15% lasted more than 2200 hours. Find the mean and standard deviation of the distribution. Solution: Let X be the random variable “life of bulb (hrs)” So, We know that and So, and

z1 is negative Using the Percentage Points of the Normal Distribution table, So, and Adding:

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.

The standardising formula to change values from a random variable X into Z values is We can also use this formula to find either m or s ( or both ) provided we know, or have enough information to find, the other unknowns.

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:

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.

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

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,

Solving simultaneously: We can change the signs in the 2nd equation and add: Substitute into either equation to find m: Adding: