normal curve - PowerPoint Presentation

Slide1

normal curve

Mrs. Aldous, Mr.

Beetz

& Mr. Thauvette

DP SL MathematicsSlide2

Normal DistributionSlide3

You should be able to…

Describe the properties of a normal distribution with mean and standard deviation

Calculate normal probabilities

Find the corresponding standardized value (

z – score) given a probabilitySlide4

You should be able to…

Use

the relation

to

standardize data or to find missing parameters and/or when

given probabilities Use the GDC to find normal probabilities or standardized values.Slide5

What is normal distribution?

Mean, median, and mode

The Normal Distribution: A probability distribution where

the

mean, median, and mode are

at the

centre

of the spread.

Are these normally distributed?

heights and mass of people

IQ scores

Scores in an examination

lifetime of a batterySlide6

Notation for normal distribution

The distribution of

X

is normally distributed

with a mean of

and a variance

of .

Note the variance is often the written as the standard deviation squared.

Write down the mean and standard deviation of each of the following normal distributions.Slide7

Heights of UK Adults

Write probability distributions to describe the heights of men and women.

Does it matter if you use feet and inches, only inches, or centimeters?Slide8

Standardizing data

For the normal distribution we can standardize the data. All standardized data has a mean of 0, and a variance (and standard deviation) of 1.

0

1

-1

Each 1 unit away from the mean is a standard deviation.

The standardized values are called

z

numbers.

The area under this curve is 1.Slide9

Using a GDC for standardized data

If you are able to use a GDC for finding normal values then this is an

easy

and

quick method.

1. Draw a sketch.

2. Use DISTR >

normalcdf

(.

3. Enter lower and upper bounds. Use -1E99 or 1E99 for .

4. Leave the mean as 1 and the standard deviation as 0.

What is the probability that z is less than (or equal to) 1?

P(

z

<1.0

) = ?

1

Probability =

0.841Slide10

Using

your GDC

Use

your GDC to

find each of the

following.

1. P (

z

<1.5)

2. P (

z

>0.85)

3. P (

z

>-1.21)

4. P (

z

<-1.75)

5. P (-1<

z

<1.5)

0.933

0.198

0.887

0.0401

0.775Slide11

Standardizing real data

Before using the normal

function,

data must be standardized. The formula for this is:

Example:

The IQ of a population is distributed normally with a mean of 100 and a standard deviation of 12. Calculate the probability that a person picked at random has an IQ greater than 118.

Always draw a diagram, and shade the region you require.

100

118

1.5

0

Write down the

‘

real

’

values.

Standardize your values. The mean will always be 0. Standardize the 118.

Use your GDC to find

z>

1.5 .

Probability =

0.0668Slide12

A GDC makes this much easier.

Example

The IQ of a population is distributed normally with a mean of 100 and a standard deviation of 12. Calculate the probability that a person picked at random has an IQ greater than 118.

Always draw a diagram, shade what you want and put down the

‘

real

’

values.

Probability =

0.0668

Using a GDC with dataSlide13

Question

1

Always

draw a

diagram. It can get you method marks.

2. Scores for

a test of anxiety are

normally distributed with a mean of 58% and a variance of 225%. Find the probability that a

random student scores:

a) more than 75

%

b) less than 60%

,

c) between 48% and 62%.

1. IQs are normally distributed with a mean of 100 and a standard deviation of 14. Calculate the probability that a person picked at random

has:

a) an IQ greater than 110,

b) an IQ lower than 95,

c) an IQ between 90 and 108.

0.238

0.360

0.479

0.129

0.553

0.353Slide14

Working backwards

Example:

Scores in an

military entrance exam

are normally distributed with a mean of 50 and a standard deviation of 20. The mark for an A is to be set such that only 10% of the

candidates will

score an A.

Find the mark required to obtain an A.

Draw a diagram and show what is required.

50

x

0

z

10%=0.1

90%=0.9

Fill in the

z

values.

Shaded area is 0.1.

Unshaded area is 0.9.

Use DISTR >

invNorm

(.

?Slide15

Working backwards continued

0

1.28

zSlide16

Working backwards with a GDC

You should use your GDC

, but always draw a diagram

.

The diagram counts as “method” for method marks.

Example:

Scores in an military entrance exam are normally distributed with a mean of 50 and a standard deviation of 20. The mark for an A is to be set such that only 10% of the candidates will score an A.

Find the mark required to obtain an A. Slide17

Question

2

2. A

maths

exam scores are normally distributed with a mean of 56% and a standard deviation of 13. A C is set so that 40% of the cohort obtain a C. The C is symmetrical about the mean such that the lower mark us 56-

a

, and the upper mark is 56+

a

. Find the bounds within which a C is given.

A

lways

draw a diagram

then use

your

GDC.

1. IQs are normally distributed with a mean of 100 and a standard deviation of 14. Only 1% of the

population are

classed as

‘

genius

’

. Find the IQ of a genius.Slide18

Starter:Slide19

You have been given a normal probability distribution sketch. Make up a question that corresponds to the sketch. You may have to provide some extra information.Please do not write on the cards.Pair up with another student and solve each other’s questions.

Continue by meeting other students.Slide20
Slide21

Quiz-quiz-tradeWork on the card you have been given.

Then get up and find another student. Work together on your questions, then trade.

Continue to quiz, quiz, and trade with other students.Slide22
Slide23

SolutionSlide24

SolutionSlide25

Exam QuestionSlide26

How do I approach this question?

What are the key areas from the syllabus?Slide27
Slide28

ExampleThe heights of boys at a particular school follow a normal distribution with a standard deviation of 5 cm. The probability of a boy being shorter than 153 cm is 0.705.

(a)

Calculate

the mean height of the boys.Slide29

Example continued…(a)

Calculate

the mean height of the boys.Slide30

Example continued…(b)

Write down

the probability of a boy being taller than 156 cm.Slide31

You should know…The normal distribution is an example of a continuous probability distribution

We write to refer to a random variable that is normally distributed with parameters and , where is the mean of the data and is the varianceSlide32

You should know…

The normal curve has the following properties:

Bell-shaped, as most of the data are clustered about the mean

Reaches its maximum height at the mean

Mean, median and mode are all equalCurve is symmetrical about the mean

Area under the normal curve represents probability, so the total area under the curve is 1Slide33

You should know…

Normally distributed data can be standardized

using the relation , and the result can

be compared to the standard normal distribution

with mean of 0 and standard deviation of 1

The z – score or standard score gives the number of standard deviations from the meanYou can use your GDC to find probabilities and values with or without standardizing firstSlide34

Be prepared…Do not confuse probabilities with

z

– scores when using the standardizing relation.

When solving problems, use a sketch or a normal curve with a shaded area indicating the probability to be given.

Problem can often be solved using the symmetry of the normal curve.Slide35

Binomial DistributionsSlide36

You should be able to…

Recognize and describe a binomial experiment

Determine the probability distribution of a binomial experiment

Calculate the probability of

r

successes in n trialsCalculate cumulative binomial probabilitiesCalculate the mean (expected value) and variance of a binomial distributionSlide37

The binomial distribution

A binomial distribution is one where there are only two distinct outcomes. Which of the following are binomial?

Binomial

Points scored when a shot is taken in basketball.

Rolling a die.

Student scores in a test.

Scoring a goal in a penalty shoot-out?

Tossing a coin.

Picking a female student from a group of students.Slide38

Binomial distributions

A die is rolled and a

‘

success

’

is noted as obtaining a square number. The process is repeated 3 times.

a) Calculate the probability of obtaining a square when a die is rolled.

b) Calculate the probability of obtaining 3 square numbers.

c) Calculate the probability of obtaining 0 square numbers.

d) Calculate the probability of obtaining 1 square number.Slide39

Pascal revisited

A die is rolled and a

‘

success

’

is noted as obtaining a square number. The process is repeated 5 times. Write down the

number of ways

of obtaining,

a) 0 squares,

b) 1 square,

c) 2 squares,

d) 3 squares,

e) 4 squares,

f) 5 squares,

FFFFF

1

SFFFF, FSFFF, FFSFF, FFFSF, FFFFS

5

SSFFF, SFSFF, SFFSF, SFFFS, FSSFF,

FSFSF, FSFFS, FFSSF, FFSFS, FFFSS

10

SSSFF, SSFSF, SSFFS, SFSSF, SFSFS,

SFFSS, FSSSF, FSSFS, FSFSS, FFSSS

10

SSSSF, SSSFS, SSFSS, SFSSS, FSSSS

5

1

SSSSS

Do you recognize the pattern in the yellow boxes?Slide40

Pascal

’

s triangle

A die is rolled and a

‘

success

’

is noted as obtaining a square number. The process is repeated 5 times. Write down the

number of ways

of obtaining,

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

5 trials

Using a GDC:Slide41

5 trials

A die is rolled and a

‘

success

’

is noted as obtaining a square number. The process is repeated 5 times. Find the probability of obtaining,

a) 0 squares,

b) 2 squares,

c) 4 squares,

2 squares from 5

probability of 2 squares

probability 3 non-squaresSlide42

10 trials

A die is rolled and a

‘

success

’

is noted as obtaining a square number. The process is repeated 10 times. Find the probability of obtaining,

a) 0 squares,

b) 2 squares,

c) 5 squares,

d) at least 2 squares.Slide43

Notation

A die is rolled and a

“

success

”

is noted as obtaining a square number. The process is repeated 10 times.

We can write this as:

number of trials,

n

probability of a success,

p

In general:

The expected result (mean) is denoted by:

E

(x)=npSlide44

Questions

1. The probability of seeing a

gecko on a given

day is known to be

0.6, and independent.

A man walks each day for a week. Find the probability that he sees a

gecko:

a) on 3 separate days,

b) on exactly 5 days,

c) at least 1 day.

2. A random variable is distributed binomially such that,

a) find the expected value of

x

.

b) find the probability that

x=

3.

c) find the probability that

x>

2.

0.194

0.261

0.998

0.104

0.135Slide45
Slide46

Probability

Tarsia

Puzzle

You need the information sheet and a set of

24 triangles

.Slide47

Example

A factory makes calculators. Over a long period, 2% of them are found to be faulty. A random sample of 100 calculators is tested.

(a)

Write down

the expected number of faulty calculators in the sample.

(b) Find the probability that three calculators are faulty.Slide48

Example continued…(b)

Find

the probability that three calculators are faulty.Slide49

Example continued…(c)

Find

the probability that more than three calculators are faulty.Slide50

Example continued…Using the GDC

(c)

Find

the probability that more than three calculators are faulty.Slide51

How do I approach this question?

What are the key area from the syllabus?

(a)

(b) Find the complement ofSlide52

You should know…A binomial experiment is one in which there are

n

independent trials. For each trial, there are only two outcomes: a success and a failure. For example, tossing a coin 10 times, consider heads success and tails failure

We write to refer to a random variable of a binomial experiment with

n independent trials and probability of a success,

pSlide53

You should know…

The probability of

r

successes in

n trials is given by where 1 – p

is the probability of a failureThe mean of a binomial distribution is given byThe variance of a binomial distribution is given bySlide54

Be prepared…Remember, when calculating a binomial probability, don’t forget that, in order for there to be exactly

r

successes, there must also be

n

– r failures. The (1 – p)

n – r factor must not be omitted.When finding cumulative probabilities less than a number don’t forget to include P(X = 0) in your calculation, that is,