Teach A Level Maths Statistics 1 Finding the

Finding the Normal Mean and Variance Statistics 1 AQA Normal Distribution diagrams in this

Finding the Normal Mean and Variance The standardising formula to change values from a

Finding the Normal Mean and Variance e. g. 1 Find the values of m

Finding the Normal Mean and Variance Tip: It’s easy to make a mistake and

Finding the Normal Mean and Variance e. g. 2 Find the values of the

Finding the Normal Mean and Variance Exercise 1. Find the values of the unknowns

Finding the Normal Mean and Variance Solution: (a) and where Using the Percentage Points

Solution: Finding the Normal Mean and Variance (b) and where Using the Percentage Points

Finding the Normal Mean and Variance In the next two examples m and s

Finding the Normal Mean and Variance e. g. 3 Find m and s if

Finding the Normal Mean and Variance and We must solve simultaneously: We can Adding:

Finding the Normal Mean and Variance e. g. 4 The lengths of a batch

Finding the Normal Mean and Variance SUMMARY Ø To find one parameter ( mean

Finding the Normal Mean and Variance Exercise 1. Find the values of m and

Finding the Normal Mean and Variance 1. and Solution: We have and Using the

Finding the Normal Mean and Variance and Solving simultaneously: We can change the signs

Finding the Normal Mean and Variance 2. A large sample of light bulbs from

Finding the Normal Mean and Variance z 1 is negative Using the Percentage Points

Finding the Normal Mean and Variance The following slides contain repeats of information on

Finding the Normal Mean and Variance e. g. 1 Find the values of the

Finding the Normal Mean and Variance e. g. 3 The lengths of a batch

Finding the Normal Mean and Variance Solving simultaneously: We can change the signs in

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

Finding the Normal Mean and Variance Statistics 1 AQA 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 Jupiter. Images and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from Jupiter. Images"

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 m or s ( or both ) provided we know, or have enough information to find, the other unknowns.

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

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.

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

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

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

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

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

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

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

Finding the Normal Mean and Variance 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, P ( Z < z 1 ) = 0 × 05

Finding the Normal Mean and Variance 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: Adding: We can change the signs in the 2 nd equation and add: Substitute into either equation to find m:

Finding the Normal Mean and Variance SUMMARY Ø 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. Ø To find both the mean and standard deviation we need to know two “pairs” of values of the variable and corresponding percentages or probabilities.

Finding the Normal Mean and Variance Exercise 1. Find the values of m and s if and 2. and 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.

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

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

Finding the Normal Mean and Variance 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. Solution: Let X be the random variable “life of bulb (hrs)” So, We know that So, and

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

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.

Finding the Normal Mean and Variance e. g. 1 Find the values of the unknowns in the following: (a) 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.

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

Finding the Normal Mean and Variance 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. Solution: Let X be the random variable “length of rod (cm)” and So, P ( Z < z 1 ) = 0 × 05

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