Transcript File - DP Mathematics SL

NORMAL CURVE
Mrs. Aldous, Mr. Beetz & Mr. Thauvette
DP SL Mathematics
Normal Distribution
You should be able to…



Describe the properties of a normal distribution with
mean mand standard deviation
s
Calculate normal probabilities
Find the corresponding standardized value
(z – score) given a probability
You should be able to…

Use the relation
z=
x-u
s
to standardize data
or to find missing parameters
given probabilities

m and/or s when
Use the GDC to find normal probabilities or
standardized values.
What is normal distribution?
The Normal Distribution:
A probability distribution
where the mean,
median, and mode are
at the centre of the
spread.
Mean, median, and mode
Are these normally
distributed?
heights and mass of people
IQ scores
Scores in an examination
lifetime of a battery
Notation for normal distribution
X ~ N (m,s
2
)
The distribution of X is normally distributed with a mean of
2
and a variance of s .
m
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.
1. X ~ N ( 30,4 2 )
m = 30,s = 4
2. X ~ N (85,36)
m = 85,s = 6
3. X ~ N (0,1)
m = 0,s =1
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?
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.
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.
-1
0
1
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.
What is the probability
2. Use DISTR >
that z is less than (or
normalcdf(.
equal to) 1?
3. Enter lower and
P(z<1.0) = ?
upper bounds. Use 1E99 or 1E99 for
.
4. Leave the mean as 1
and the standard
deviation as 0.
±¥
Probability = 0.841
1
Using your GDC
Use your GDC to find each of the following.
1. P (z<1.5)
0.933
2. P (z>0.85)
0.198
3. P (z>-1.21)
0.887
4. P (z<-1.75)
0.0401
5. P (-1<z<1.5)
0.775
Standardizing real data
Before using the normal function, data must be standardized. The
formula for this is:
x -m
=z
s
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.
X ~ N 100,12 2
Always draw a diagram, and shade
the region you require.
(
)
Write down the ‘real’ values.
Standardize your values. The mean
will always be 0. Standardize the
118.
118 -100
= 1.5
12
Use your GDC to find z>1.5 .
100
118
0
1.5
Probability = 0.0668
Using a GDC with data
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.
A GDC makes this much easier.
Always draw a diagram, shade
what you want and put down the
‘real’ values.
X ~ N (100,12 2 )
Probability = 0.0668
Question 1
Always draw a diagram. It can get you method marks.
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,
0.238
b) an IQ lower than 95,
0.360
c) an IQ between 90 and 108.
0.479
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:
0.129
a) more than 75%
b) less than 60%,
c) between 48% and 62%.
0.553
0.353
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.
Fill in the z values.
X ~ N (50,20 2 )
90%=0.9
10%=0.1
Shaded area is 0.1.
Unshaded area is 0.9.
Use DISTR > invNorm(.
50
0
x
?
z
Working backwards continued
x -m
s
0
1.28
=z
x - 50
= 1.28
20
x = 1.28 ´ 20 + 50
xx == 76%
75.6%
z
Working backwards with a GDC
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.
You should use your GDC, but always draw a diagram.
The diagram counts as “method” for method marks.
x = 75.6%
Question 2
Always 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.
x »133
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.
49.2 £ x £ 62.8
Starter:




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.
Quiz-quiz-trade



Work 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.
Solution
Solution
Exam Question


How do I approach this question?
What are the key areas from the syllabus?
Example
The 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.
Example continued…
(a) Calculate the mean height of the boys.
Example continued…
(b) Write down the probability of a boy being taller
than 156 cm.
You should know…


The normal distribution is an example of a
continuous probability distribution
(
)
We write X ~ N m,s 2 to refer to a random
variable that is normally distributed with
parameters m and 2, where m is the mean of the
data and 2 is the variance
s
s
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 equal
 Curve is symmetrical about the mean
 Area under the normal curve represents
probability, so the total area under the curve is
1
You should know…

Normally distributed data can be standardized
using the relation
z=
x-u
s
, 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 mean
 You can use your GDC to find probabilities and
values with or without standardizing first
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.
Binomial Distributions
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 trials
Calculate cumulative binomial probabilities
Calculate the mean (expected value) and variance
of a binomial distribution
The binomial distribution
A binomial distribution is one where there are only two distinct
outcomes. Which of the following are binomial?
Picking a
female student
from a group of
students.
Student scores
in a test.
Scoring a goal in a
penalty shoot-out?
Binomial
Points scored
when a shot is
taken in
basketball.
Tossing a coin.
Rolling a die.
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.
1
3
b) Calculate the probability
of obtaining 3 square
numbers.
æ 1 ö3 1
ç ÷ =
è 3 ø 27
c) Calculate the probability
of obtaining 0 square
numbers.
æ 2 ö3 8
ç ÷ =
è 3 ø 27
d) Calculate the probability
of obtaining 1 square
number.
æ 1 2 2ö æ 2 1 2ö æ 2 2 1ö
ç ´ ´ ÷+ç ´ ´ ÷+ç ´ ´ ÷
è 3 3 3ø è 3 3 3ø è 3 3 3ø
æ 1 2 2ö
= 3ç ´ ´ ÷
è 3 3 3ø
12
=
27
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, FFFFF
1
b) 1 square,
5
SFFFF, FSFFF, FFSFF, FFFSF, FFFFS
c) 2 squares, SSFFF, SFSFF, SFFSF, SFFFS, FSSFF,
FSFSF, FSFFS, FFSSF, FFSFS, FFFSS
10
d) 3 squares, SSSFF, SSFSF, SSFFS, SFSSF, SFSFS,
SFFSS, FSSSF, FSSFS, FSFSS, FFSSS
10
e) 4 squares, SSSSF, SSSFS, SSFSS, SFSSS, FSSSS
5
f) 5 squares,
SSSSS
Do you recognize the pattern in the yellow boxes?
1
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
1
1
1
1
2
3
4
5
5 trials
6
Using a GDC:
1
3
1
6
4
1
10
10
5
1
15
20 15 6
5
C2
1
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,
2 squares
from 5
æ 2 ö 5 32
ç ÷ =
è 3 ø 243
2
3
æ
ö
æ
ö
1 2
5
C
ç
( 2 )è 3÷ø çè 3÷ø
probability of
2 squares
c) 4 squares,
1 8
= 10 ´ ´
9 27
80
=
243
probability 3
non-squares
4
1
æ
ö
æ
ö
1
2
10
5
( C4 )çè 3÷ø çè 3÷ø = 243
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,
æ 2 ö10
ç ÷ = 0.0173
è 3ø
b) 2 squares,
2
8
æ
ö
æ
ö
1
2
(10C2 )çè 3÷ø çè 3÷ø = 0.195
c) 5 squares,
5
5
æ
ö
æ
ö
1
2
(10C5 )çè 3÷ø çè 3÷ø = 0.137
d) at least 2 squares.
1- [Pr(0) + Pr(1)] = 0.896
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:
æ 1ö
X ~ Bç10, ÷
è 3ø
number of trials, n
In general:
probability of a success, p
X ~ B(n, p)
The expected result (mean) is denoted by:
E(x)=np
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,
0.194
b) on exactly 5 days,
0.261
c) at least 1 day.
0.998
æ 1ö
è 6ø
2. A random variable is distributed binomially such that, X ~ Bç8, ÷
a) find the expected value of x.
b) find the probability that x=3.
c) find the probability that x>2.
8
6
0.104
0.135
Probability Tarsia Puzzle
You need the information sheet and a set of 24 triangles.
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.
Example continued…
(b) Find the probability that three calculators are
faulty.
Example continued…
(c) Find the probability that more than three
calculators are faulty.
Example continued…
Using the GDC
(c) Find the probability that more than three
calculators are faulty.
How do I approach this question?
 What are the key area from the syllabus?
(a)

(b) Find the complement of
P ( X £ 1)
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 X ~ B ( n, p ) to refer to a random
variable of a binomial experiment with n
independent trials and probability of a success, p
You should know…

The probability of r successes in n trials is given by
æ nö r
n-r
P ( X = r ) = ç ÷ p (1- p )
èrø
where 1 – p is the probability of a failure

The mean of a binomial distribution is given by

The variance of a binomial distribution is given by
E ( X ) = m = np
Var ( X ) = npq
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,