Transcript S5_Int2_Outcome 1 ch

Int2
Unit 3 Outcome 1
Algebraic Operations
Simplify
Adding and Subtracting fractions
Multiply and Divide fractions
Change the subject of the formula
Harder
Harder
Harder
Relative Frequency & Probability
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1
Unit 3 Outcome 1
Int2
Starter Questions
1.
Simplify the following fractions :
(a)
3a²b4
9b²
= a² b2
3
(b) 4x² + x
= x (4x +1) = x
4x + 1
(4x +1)
2.
Calculate
3.
Calculate
3√8
4.
Calculate
1 + 1 = 1 x2 = 2 + 1 = 3 = 1
3
6 3 2
6
6 6
2
√16
= 4
(4 x 4)
= 2
(2 x 2 x 2)
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Algebraic Operations
Int2
Learning Intention
Success Criteria
1. To simplify algebraic
fractions.
1. Know the term quartiles.
2. Calculate quartiles given a
frequency table.
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Int2
Unit 3 Outcome 1
Adding Algebraic Fractions
Example 1b
Example 1a
1 + 4
7
7
Common
denominator
4 + 2
f
f
LCM = 7
LCM = f
= 1 + 4
7
= 4 + 2
f
=
5
7
The letter
or number is
the same on
the bottom
line
= 6
f
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Int2
Unit 3 Outcome 1
Adding Algebraic Fractions
Example 2b
Example 2a
4 + 5
11 11
Common
denominator
8 + 5
w
w
LCM = 11
LCM = w
= 4 + 5
11
= 8 + 5
w
=
9
11
The letter
or number is
the same on
the bottom
line
= 13
w
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5
Int2
Unit 3 Outcome 1
Adding Algebraic Fractions
When the denominator is different
Example 2a
4
5
+
5
10
Common
denominator
Cross multiply
=4 x10 + 5 x5
5 x10
10x5
=
8
10
+
5 = 13
10 10
The letter
or number is
the same on
the bottom
line
Example 2b
4 + 5
c
w
Cross multiply
=4 x w + 5 x c
c x w
w x c
= 4w +
cw
5c
wc
= 4w + 5c
cw
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Int2
Unit 3 Outcome 1
Subtracting Algebraic Fractions
Compare with real numbers to help
Example 3b
Example 3a
6 - 2
9
9
LCM = 7
= 6 - 2
9
=
4
9
The letter or
number is the
same on the
bottom Line
Common
denominator
7 - 4
d
d
LCM = f
= 7 - 4
d
= 3
d
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Int2
Unit 3 Outcome 1
Subtracting Algebraic Fractions
Example 2b
Example 2a
6 - 2
9
9
LCM = 7
= 6 - 2
9
=
4
9
The letter or
number is the
same on the
bottom Line
Common
denominator
7 - 4
d
d
LCM = f
= 7 - 4
d
= 3
d
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Statistics
Int2
Quartiles from Frequency Tables
To find the quartiles of an ordered list you consider its
length. You need to find three numbers which break the
list into four smaller list of equal length.
Example 1 : For a list of 24 numbers, 24 ÷ 6 = 4
6 number
Q1
6 number
Q2
6 number
Q3
R0
6 number
The quartiles fall in the gaps between
Q1 : the 6th and 7th numbers
Q2 : the 12th and 13th numbers
Q3 : the 18th and 19th numbers.
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Int2
Statistics
Quartiles from Frequency Tables
Example 2 : For a list of 25 numbers, 25 ÷ 4 = 6
6 number Q1
6 number
1 No.
6 number
R1
Q3 6 number
Q2
The quartiles fall in the gaps between
Q1 : the 6th and 7th
Q2 : the 13th
Q3 : the 19th and 20th numbers.
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Statistics
Int2
Quartiles from Frequency Tables
Example 3 : For a list of 26 numbers, 26 ÷ 4 = 6
6 number
1 No.
Q1
6 number
Q2
6 number
1 No.
R2
6 number
Q3
The quartiles fall in the gaps between
Q1 : the 7th number
Q2 : the 13th and 14th number
Q3 : the 20th number.
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Statistics
Int2
Quartiles from Frequency Tables
Example 4 : For a list of 27 numbers, 27 ÷ 4 = 6
6 number
1 No.
Q1
6 number
1 No.
Q2
6 number
1 No.
R3
6 number
Q3
The quartiles fall in the gaps between
Q1 : the 7th number
Q2 : the 14th number
Q3 : the 21th number.
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Statistics
Int2
Quartiles from Frequency Tables
Example 4 : For a ordered list of 34.
Describe the quartiles.
34 ÷ 4 = 8
8 number
1 No.
Q1
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8 number
R2
Q2
8 number
1 No.
8 number
Q3
The quartiles fall in the gaps between
Q1 : the 9th number
Q2 : the 17th and 18th number
Q3 : the 26th number.
13
Statistics
Int2
Quartiles from Frequency Tables
Now try Exercise 1
Start at 1b
Ch11 (page 162)
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Starter Questions
Int2
1. Multiply out the brackets and simplify
4(y + 3) - 3(8 - x)
2. Find the gradient and the y - intercept
for the line with equation 2y = - 4x + 10
3. Find the quartiles for the ordered 6 numbers
10, 12, 14, 18, 22, 30,32
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Statistics
Int2
Quartiles from Cumulative Frequency Table
Learning Intention
1. To explain how to
calculate quartiles from
Cumulative Frequency
Table.
Success Criteria
1. Find the quartile values from
Cumulative Frequency Table.
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Int2
Statistics
Quartiles from Cumulative Frequency Table
Example 1 :
The frequency table shows the length
of phone calls ( in minutes) made from
an office in one day.
Time
Freq.
(f)
Cum. Freq.
1
2
2
2
3
5
3
5
10
4
8
18
5
4
22
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Statistics
Int2
Quartiles from Cumulative Frequency Table
We use a combination of quartiles from a frequency table
and the Cumulative Frequency Column.
For a list of 22 numbers, 22 ÷ 4 = 5
5 number
1 No.
Q1
5 number
Q2
5 number
1 No.
R2
5 number
Q3
The quartiles fall in the gaps between
Q1 : the 6th number Q1 : 3 minutes
Q2 : the 11th and 12th number Q2 : 4 minutes
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Q3 :
the 17th number. Q3 : 4 minutes
18
Statistics
Int2
Quartiles from Cumulative Frequency Table
Example 2 :
A selection of schools were asked
how many 5th year sections they have.
Opposite is a table of the results.
Calculate the quartiles for the results.
No. Of
Sections
Freq. Cum. Freq.
(f)
4
3
3
5
5
8
6
8
16
7
9
25
8
8
33
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Statistics
Int2
Quartiles from Cumulative Frequency Table
We use a combination of quartiles from a frequency table
and the Cumulative Frequency Column.
Example 2 : For a list of 33 numbers, 33 ÷ 4 = 8
8 number Q1
8 number
1 No.
8 number
r1
Q3 8 number
Q2
The quartiles fall in the gaps between
Q1 : the 8th and 9th numbers Q1 : 5.5
Q2 : the 17th number Q2 : 7
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Q3 :
the 25th ad 26th numbers. Q3 : 7.5
20
Int2
Statistics
Quartiles from Cumulative Frequency Table
Now try Exercise 2
Ch11 (page 163)
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Starter Questions
Int2
1. Find the area of the triangle.
3cm
2cm
29o
4cm
2. Write down the two conditions
for using the cosine rule.
C
70o
3. Find the length of AC.
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A
53o
B
8cm
22
Quartiles from
Cumulative Frequency
Graphs
Int2
Learning Intention
1. To show how to estimate
quartiles from cumulative
frequency graphs.
Success Criteria
1. Know the terms quartiles.
2. Estimate quartiles from
cumulative frequency graphs.
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Int2
Quartiles from
Cumulative Frequency
Graphs
Number of
sockets
10
20
30
40
50
60
Cumulative
Frequency
2
9
24
34
39
40
24
New Term
Cumulative Frequency
Interquartile range
Graphs range
Semi-interquartile
Int2
(Q3 – Q1 )÷2 = (36 - 21)÷2
=7.5
Cumulative Frequency
45
Quartiles
40
Q3
35
30
Q2
25
Q3 =36
Q2 =27
20
Q1
15
40 ÷ 4 =10
Q1 =21
10
5
0
0
10
20
30
40
50
Number of Sockets
60
70
25
Int2
Quartiles from
Cumulative Frequency
Graphs
Km travelled on
1 gallon (mpg)
20
25
30
35
40
45
50
Cumulative
Frequency
3
11
30
53
69
76
80
26
New Term
Cumulative Frequency
Interquartile range
Graphs range
Semi-interquartile
Cumulative Frequency
Int2
(Q3 – Q1 )÷2 = (37 - 28)÷2
=4.5
90
80
Q3 = 37
70
60
Quartiles
80 ÷ 4 =20
Q2 = 32
50
40
Q1 =28
30
20
10
0
0
10
20
30
40
50
60
Km travelled on 1 gallon (mpg)
27
Quartiles from
Cumulative Frequency
Graphs
Int2
Now try Exercise 3
Ch11 (page 166)
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Starter Questions
Int2
1. Factorise x2  11x  28
2. Find the volume of a cone 15cm in height
and 10cm in diameter.
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Standard Deviation
Int2
Learning Intention
1. To explain the term and
calculate the Standard
Deviation for a collection
of data.
Success Criteria
1. Know the term Standard
Deviation.
1. Calculate the Standard
Deviation for a collection of
data.
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Standard Deviation
For a FULL set of Data
Int2
The range measures spread. Unfortunately any big
change in either the largest value or smallest score
will mean a big change in the range, even though only
one number may have changed.
The semi-interquartile range is less sensitive to a single
number changing but again it is only really based on two
of the score.
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Standard Deviation
For a FULL set of Data
Int2
A measure of spread which uses all the data is the
Standard Deviation
The deviation of a score is how much the score differs
from the mean.
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Int2
Step 5 :
Deviation
Step 2 : Score - Mean
Step 1 :Standard
Find the mean
Step 4 : Mean
square deviation
2
the square
rootData
of step 4
aTake
FULL
set of
Step For
3 : (Deviation)
375 ÷ 5 = 75
68 ÷ 5 = 13.6
√13.6 = 3.7
Example 1 : Find the standard deviation of these five
scores 70,
72, 75, 78,
80.(to 1d.p.)
Standard
Deviation
is 3.7
Score
70
72
75
78
80
Totals
375
Deviation
(Deviation)2
-5
-3
0
3
5
0
25
9
0
9
25
68
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5Deviation
: square deviation
Step 1 : FindStandard
the
mean
Step
4Step
: Mean
Step
2 : Score - Mean
Int2
For
a
FULL
set
of
Data
2
Take the square root of step 4
Step180
3 : ÷(Deviation)
6 = 30
962 ÷ 6 = 160.33
= 12.7
(to 1d.p.)
Example 2 √160.33
: Find the
standard
deviation of these six
amounts of money £12, £18, £27, £36, £37, £50.
Standard Deviation is £12.70
Score
Deviation
(Deviation)2
12
-18
324
18
-12
144
27
36
37
Totals
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50
180
-3
6
7
20
0
9
36
49
400
962
34
Standard Deviation
For a FULL set of Data
Int2
When Standard Deviation
is LOW it means the data
values are close to the
MEAN.
Mean
When Standard Deviation
is HIGH it means the data
values are spread out from
the MEAN.
Mean
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Standard Deviation
Int2
Now try Exercise 4
Ch11 (page 169)
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Starter Questions
Int2
1. Construct a cumulative frequency table
For the data below.
Waist Sizes
Frequency
28”
7
30”
12
32”
23
34”
14
2. Solve the simultaneous equations
x  y  10
and
2x - y  5
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Standard Deviation
For a Sample of Data
Int2
Learning Intention
1. To show how to calculate
the Standard deviation
for a sample of data.
Success Criteria
1. Construct a table to calculate
the Standard Deviation for a
sample of data.
2. Use the table of values to
calculate Standard Deviation
of a sample of data.
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Int2
Standard Deviation
For a Sample ofWe
Data
will use this
version because it is
easier
to use
in a sample
In real life situations it is normal
to work
with
practice ).
!
of data ( survey / questionnaire
We can use two formulae to calculate the sample deviation.
s
 ( x  x)
2
n 1
s = standard deviation
x = sample mean
 x 
2
s
x



n 1
n
∑ = The sum of
n = number in sample
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2
2:
Q1a. Calculate the mean : Q1a.Step
Calculate
the
StandardStep
Deviation
3 :sample deviation
Step592
1 : ÷ 8 = 74
all the values
For a SampleSquare
of Data
Int2
and find the total
Sum all the valuesUse formula to calculate
sample have
deviation
Example 1a : Eight athletes
heart rates
70, 72, 73, 74, 75, 76, 76 and 76.
s
s
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
2
x


x



n 1
 43842
Heart rate (x)
2
8 1
8
2
4900
72
 43842   43808
5184
73
7
s
n
592 


70
x2
5329
74
5476
75
5625
76
s  4.875
5776
s76 2.2 (to 1 d . p5776
.)
76
5776
Totals ∑x = 592
40
∑x2 = 43842
Int2
Standard
Deviation
Q1b(i) Calculate the mean : Q1b(ii) Calculate the
sample
deviation
For
Sample of
Data
720
÷8a
= 90
Example 1b : Eight office staff train as athletes.
Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM
s
s
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
x
2

x



n 1
 65218
2
s 81
83
720 


8 1
80
n
8
Heart rate (x)
90
2
94
x2
6400
 65218  64800
7
s  418
6561
6889
8100
8836
96
9216
100
10000
Totals ∑x = 720
41
∑x2 = 65218
s96 20.4 (to 1d.9216
p.)
Standard
Q1b(iii) WhoDeviation
are fitter
Q1b(iv) What does the
athletes
or of
staff.
Forthe
adeviation
Sample
Data
tell us.
Int2
Compare means
Staff data is more spread
Athletes are fitter
out.
Athletes
Staff
Mean  74 BPM
Mean  90 BPM
s  2.2 (to 1d. p.)
s  20.4 (to 1d. p.)
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Standard Deviation
For a Sample of Data
Int2
Now try Ex 5 & 6
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Starter Questions
Int2
1. If lines have the same gradient
What is special about them.
2. Factorise x2 + 8x + 15
33o
3. Find the missing angles.
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Scatter Graphs
Int2
Construction of Scatter Graphs
Learning Intention
1. To construct and interpret
Scattergraphs.
Success Criteria
1. Construct and understand
the Key-Points of a
scattergraph.
2. Know the term positive and
negative correlation.
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This scattergraph
shows the heights
and weights of a
Int2 sevens football team
Write down height and
Scatter Graphs
weight of each player.
Construction of Scatter Graph
Team
180
Tim
160
Height (cm)
Bob
Sam
140
Joe
Gary
Jim
Dave
120
100
0
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20
40
60
Weight (kg)
46
Scatter Graphs
Construction of Scatter Graph
Int2
When two quantities are strongly connected we say there is a
strong correlation between them.
Best fit line
x x
x x
x x
Strong positive
correlation
x
x x
x
x
x
Best fit line
Strong negative
correlation
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Scatter Graphs
Int2
Construction of Scatter Graph
Key steps to:
Drawing the best fitting straight line to a scatter graph
1. Plot scatter graph.
2. Calculate mean for each variable and plot the
coordinates on the scatter graph.
3. Draw best fitting line, making sure it goes through
mean values.
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Draw in the
best fit line
Find the mean
Mean Age = 2.9
for theAge
and Prices
values.Price = £6000
Int2 Mean
Scatter Graphs
Construction of Scatter Graph
Price
Age (£1000)
9
8
2
3
3
3
4
4
5
8
7
6
5
5
4
2
Is there
a correlation?
If yes, what
kind?
10
Car prices (£1000)
1
1
12
8
6
4
2
0
0
2
4
6
8
10
12
Ages (Years)
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Scatter Graphs
Construction of Scatter Graph
Int2
Key steps to:
Finding the equation of the straight line.
1. Pick any 2 points of graph ( pick easy ones to work with).
2. Calculate the gradient using : a  y2  y1
x2  x1
3. Find were the line crosses y–axis this is b.
4. Write down equation in the form : y = ax + b
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Crosses y-axis at 10
Scatter Graphs
Int2
12
Pick points
(0,10) and (3,6)
a
10  6
 1.38
30
Car prices (£1000)
10
8
6
4
2
0
0
y = 1.38x + 10
2
4
6
8
10
12
Ages (Years)
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Scatter Graphs
Construction of Scatter Graph
Int2
Now try
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Starter Questions
Int2
1. Write the five figure summary for the data.
1, 1, 2, 3, 8, 3, 2
2. Factorise
h2 - 49
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Probability
Int2
Learning Intention
1. To understand probability
in terms of the number
line and calculate simple
probabilities.
Success Criteria
1. Understand the probability
line.
2. Calculate simply probabilities.
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Probability
Likelihood Line
Int2
0
Impossible
Seeing
a butterfly
In July
0.5
Not very
likely
School
Holidays
Evens
Winning the
Lottery
1
Very
likely
Baby Born
A Boy
Certain
Go back
in time
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Probability
Likelihood Line
Int2
0
Impossible
It will
Snow in winter
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0.5
Not very
likely
Evens
Homework Everyone getting
Every week
100 % in test
1
Very
likely
Certain
Toss a coin Going without
That land
Food
Heads
for a year.
56
Probability
Int2
We can normally attach a value
to the probability of an event happening.
To work out a probability
number of outcomes
P(A) =
Total number of possible outcomes
Probability is ALWAYS in the range 0 to 1
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Probability
Number Likelihood Line
Int2
1
0
2
0.1
Impossible
3
0.2
4
0.3
5
0.4
6
0.5
0.6
8
7
0.7
0.8
0.9
Evens
Q. What is the chance of picking a number between 1 – 8 ?
1
Certain
P=
8
=1
8
4
P(E) =
= 0.5
8
Q. What is the chance of picking the number 1 ?
1
P(1)
=
= 0.125
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58
Q. What is the chance of picking a number that is even ?
Probability
Likelihood Line
Int2
52 cards in a pack of cards
0
0.1
Impossible
0.2
0.3
Not very
likely
0.4
0.5
0.6
0.7
Evens
0.8
Very
likely
Q. What is the chance of picking a red card ?
26
P (Red) =
52
Q. What is the chance of picking a diamond ?
P (D) =
Q. What is the chance of picking ace ?
4
52
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P (Ace) =
1
0.9
Certain
= 0.5
13
= 0.25
52
= 0.08
59
Probability
Int2
Now try Ex 8
Ch11 (page 177)
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Starter Questions
Int2
1. Factorise 16x2 -36
2. The average price of a two wek holiday is £1000.
The prices depreciates @ 2% each year.
How much is the average price of a holiday
after 3 years.
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Relative Frequencies
Int2
Learning Intention
1. To understand the term
relative frequency.
Success Criteria
1. Know the term relative
frequency.
2. Calculate relative frequency
from data given.
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Relative Frequencies
Int2
Relative Frequency
Relative
Frequency
always added
up to 1
How often an event happens compared
to the total number of events.
Example : Wine sold in a shop over one week
Country
Frequency
Relative Frequency
France
180
Italy
90
90 ÷ 360 = 0.25
Spain
90
90 ÷ 360 =
Total
360
1
180 ÷ 360 = 0.5
0.25
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Relative Frequencies
Int2
Example
Calculate the relative frequency for boys and girls
born in the Royal Infirmary hospital in December 2007.
Boys Girls
Total
Frequency
300
200
500
Relative Frequency
0.6
0.4
1
Relative
Frequency
adds up
to 1
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Relative Frequencies
Int2
Now try Ex 9
Ch11 (page 179)
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Int2
Starter Questions
1. Write the five figure summary for the data below.
13, 19, 25, 25, 28, 32, 34, 36
2. The population of Scotland was 6 Million in 2000.
It increased by 3% each year for 4 years.
What is the population after the 4 years.
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Int2
Probability from Relative
Frequency
Learning Intention
1. To understand the
connection of probability
and relative frequency.
Success Criteria
1. Know the term relative
frequency.
2. Estimate probability from
the relative frequency.
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When the sum of the
Probabilityfrequencies
from Relative
is LARGE the
Int2
relative frequency is a good
Frequency
estimate of the probability
of an outcome
Example 1
Three students carry out a survey to study left
handedness in a school. Results are given below
Number of
Left - Hand
Students
Total
Asked
Sean
2
10
Karen
3
25
Daniel
20
200
Relative
Frequency
2
= 0.2
10
3
= 0.12
25
20
= 0.1
200
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Int2
Who’s
results would
you Relative
Probability
from
Megan’s
use as a estimate of the
Frequency
probability of a house
being alarmed ?
Example 2
Three
carry out a survey to study peoples
What
is students
the
favourite colours. Results
probability
0.4 are given below
that a house is
Number of
Relative
Total
alarmed ?
Alarmed
Asked Frequency
Houses
Paul
7
10
Amy
12
20
Megan
40
100
7
= 0.7
10
12
= 0.6
20
40
= 0.4
100
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Probability from
Relative Frequency
Int2
Now try Ex 10
Ch11 Start at Q2
(page 181)
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