Mrs Aldous Mr Beetz amp Mr Thauvette DP SL Mathematics Normal Distribution You should be able to Describe the properties of a normal distribution with mean and standard deviation ID: 577345
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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.Slide20Slide21
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.Slide22Slide23
SolutionSlide24
SolutionSlide25
Exam QuestionSlide26
How do I approach this question?
What are the key areas from the syllabus?Slide27Slide28
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
0.135Slide45Slide46
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,