11-3 Use Normal Distributions Algebra II

The Normal Distribution (Bell Curve) Average contents 50 Mean = μ = 50 Standard deviation = σ = 5

REMEMBER your box plot Middle 50% LQ UQ range

Normal Distribution

The normal distribution is a theoretical probability the area under the curve adds up to one

The normal distribution is a theoretical probability the area under the curve adds up to one A Normal distribution is a theoretical model of the whole population. It is perfectly symmetrical about the central value; the mean μ represented by zero.

As well as the mean the standard deviation (σ) must also be known. The X axis is divided up into deviations from the mean. Below the shaded area is one deviation from the mean. As well as the mean the standard deviation (σ) must also be known.

Two standard deviations from the mean

Three standard deviations from the mean

Fill in % for each section 0.15%/2.35%/13.5%/34%/34%/13.5%/2.35%/0.15%

A handy estimate – known as the Imperial Rule for a set of normal data: 68% of data will fall within 1σ of the μ P( -1 < z 1 ) = 0.683 68.3%

95% of data fits within 2σ of the μ P( -2 < z 2 ) = 0.954 95.4%

99.7% of data fits within 3σ of the μ P( -3 < z 3 ) = 0.997 99.7%

Ex. 1 ) guided practice A.) P(x ≤ x¯) B.) P(x ≥ x¯) C.) P(x¯ ≤ x ≤ x¯ + 2Ơ) D.) P ( x¯-Ơ ≤ x ≤ x¯ ) E.) P ( x ≤ x¯ -3Ơ ) F.) P( x ≥ x¯ + Ơ)

Interpret normally distributed data (when given mean and standard deviation) Ex. 2) The score on an exam for entrance to a firefighter program are normally distributed with a mean of 200 points and standard deviation of 20 points. A.) About what percent of the candidates score lower than 160 points? B.) Candidates with scores above 220 are admitted into the training program. About what percent of the candidates are accepted into the program?

11-3B Z-score  

Ex. 1 A P(x ≤ 90) Scientists conducted aerial surveys of a seal sanctuary & recorded the number of x seals they observed during each survey. The number of seals observed were normally distributed with a mean of 73 seals & a standard deviation of 14.1 seals. Find the probability that at most 90 seals were observed during a survey. Use the guided practice #8 on pg. 759

Ex. 1B If Ex. 1 said at least 90 or P(x≥90), then 1 - _____ (decimal)

Ex. 2 Given mean = 64 standard dev. = 7 Find P(50 ≤ x ≤ 70)

Simple problems solved using the imperial rule - firstly, make a table out of the rule <-3 -3 to -2 -2 to -1 -1 to 0 0 to 1 1 to 2 2 to 3 >3 0% 2% 14% 34% The heights of students at a college were found to follow a bell-shaped distribution with μ of 165cm and σ of 8 cm. What proportion of students are smaller than 157 cm 16%

Simple problems solved using the imperial rule - firstly, make a table out of the rule <-3 -3 to -2 -2 to -1 -1 to 0 0 to 1 1 to 2 2 to 3 >3 0% 2% 14% 34% The heights of students at a college were found to follow a bell-shaped distribution with μ of 165cm and σ of 8 cm. Above roughly what height are the tallest 2% of the students? 165 + 2 x 8 = 181 cm

Task – class 10 minutes finish for homework Exercise A Page 76

Assignment

Sometimes you will see it using phi =. The Bell shape curve happens so when recording continuous random variables that an equation is used to model the shape exactly. Put it into your calculator and use the graph function. Sometimes you will see it using phi =.

There are normal distribution tables Luckily you don’t have to use the equation each time and you don’t have to integrate it every time you need to work out the area under the curve – the normal distribution probability There are normal distribution tables

How to read the Normal distribution table Φ(z) means the area under the curve on the left of z

How to read the Normal distribution table Φ(0.24) means the area under the curve on the left of 0.24 and is this value here:

Values of Φ(z) Φ(-1.5)=1- Φ(1.5)

Values of Φ(z) Φ(0.8)=0.78814 (this is for the left) Area = 1-0.78814 = 0.21186

Values of Φ(z) Φ(1.5)=0.93319 Φ(-1.00) Shaded area = =1- Φ(1.00) =1-0.84134 =0.15866 Shaded area = Φ(1.5)- Φ(-1.00) = 0.93319 - 0.15866 = 0.77453

Task Exercise B page 79

Solving Problems using the tables NORMAL DISTRIBUTION The area under the curve is the probability of getting less than the z score. The total area is 1. The tables give the probability for z-scores in the distribution X~N(0,1), that is mean =0, s.d. = 1. ALWAYS SKETCH A DIAGRAM Read the question carefully and shade the area you want to find. If the shaded area is more than half then you can read the probability directly from the table, if it is less than half, then you need to subtract it from 1. NB If your z-score is negative then you would look up the positive from the table. The rule for the shaded area is the same as above: more than half – read from the table, less than half subtract the reading from 1.

You will have to standardise if the mean is not zero and the standard deviation is not one

Task Exercise C page 168

Normal distribution problems in reverse Percentage points table on page 155 Work through examples on page 84 and do questions Exercise D on page 85

Key chapter points The probability distribution of a continuous random variable is represented by a curve. The area under the curve in a given interval gives the probability of the value lying in that interval. If a variable X follows a normal probability distribution, with mean μ and standard deviation σ, we write X ̴ N (μ, σ2) The variable Z= is called the standard normal variable corresponding to X

Key chapter points cont. If Z is a continuous random variable such that Z ̴ N (0, 1) then Φ(z)=P(Z<z) The percentage points table shows, for probability p, the value of z such that P(Z<z)=p