Download Units 3 and 4 Further Maths

Units 3 and 4
Further Maths
Exam Revision Lecture 1
presented by Mr Bohni
Planning
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Today is the 27th of August
Your first exam is on the 2nd of November
You have 66 more days until this exam
10 of these days will be during the holidays
5 will be during swot vac
20 will be on the weekend
And 1 will be your muck up day
This leaves only 30 more days of actual school
where your teachers are available to help you
DON’T WASTE TIME
• Of the 30 days that you have left of actual school,
you are likely to only have 21 lessons of further
maths.
• There are 2 past exams every year from 2002
onwards and your teachers have an additional 12
practice papers
• That means that there are 24 past exams that you
need to do before the exam
This means that you should be completing more
than 1 exam per lesson... Ask yourself are you
capable of doing this yet?
The Exams
• There are 2 exams for further maths
– Exam 1 is a multiple choice Exam
– Exam 2 is an extended response exam
• Both exams will be in 2 sections, the core section
(Section A) and a module section (Section B).
• All students must answer the questions in Section
A and in Section B you can answer questions from
any 3 of the 5 modules.
• Your CAS Calculators are allowed in both exams
• You are allowed 1 bound reference in both exams
Section A - The Core
• The core section of the further maths course is
about Univariate Data, Bivariate Data, Linear
Regression and Time series.
• These questions are compulsory for all
students to complete. DO NOT SKIP THE CORE
SECTION!!!
So what’s important in the core?
Be familiar with the
different types of
data
– Discrete
– Continuous
– Nominal
– Ordinal
Know your graphs and tables
Be familiar with stem and leaf plots, frequency
tables, frequency histograms, bar charts,
column charts, dot plots, and all those other
wonderful things.
5 summary statistics
What is the minimum snow depth?
What is the median snow depth?
What is the maximum snow depth?
What was that dot?
The dot was an outlier, a value that didn’t
seem to fit in with the rest of the data.
Perhaps on the day the reading was taken, the
value wasn’t recorded properly or it was just a
freak occurrence.
A data point should only fit onto the whisker
of a box plot if it falls within 1½ interquartile
ranges of Q1 or Q3.
Skewed Data
If the mean appears to be closer to one end of the data
it is said to be skewed.
– If it is closer to the right it is positively skewed
(in maths being right is a positive thing)
– If it is closer to the left it is negatively skewed
(in maths being left behind is a negative thing)
Standard Deviation
The standard deviation is a measure of how
spread out the data is
A larger standard deviation means that we
have a much larger range of data
The 68%, 95%, 99.7% rule
34%
2.35%
34%
13.5%
13.5%
99.7%
2.35%
The pulse rates of a large group of 18-year-old students are approximately normally
distributed with a mean of 75 beats/minute and a standard deviation of 11
beats/minute.
Question 6
The percentage of 18-year-old students with pulse rates less than 75 beats/minute
is closest to
A. 32%
B. 50%
C. 68%
D. 84%
E. 97.5%
Question 7
The percentage of 18-year-old students with pulse rates less than 53 beats/minute
or greater than 86 beats/minute is closest to
A. 2.5%
B. 5%
C. 16%
D. 18.5%
E. 21%
The pulse rates of a large group of 18-year-old students are approximately normally
distributed with a mean of 75 beats/minute and a standard deviation of 11
beats/minute.
Question 6
The percentage of 18-year-old students with pulse rates less than 75 beats/minute
is closest to
A. 32%
ANSWER
B. 50%
If 75 beats/minute is the mean, then it is exactly
C. 68%
in the middle of the data meaning 50% is above
D. 84%
and 50% is below, so the answer is B
E. 97.5%
Question 7
The percentage of 18-year-old students with pulse rates less than 53 beats/minute
or greater than 86 beats/minute is closest to
A. 2.5%
B. 5%
C. 16%
D. 18.5%
E. 21%
The pulse rates of a large group of 18-year-old students are approximately normally
distributed with a mean of 75 beats/minute and a standard deviation of 11
beats/minute.
Question 6
The percentage of 18-year-old students with pulse rates less than 75 beats/minute
is closest to
A. 32%
ANSWER
B. 50%
If 75 beats/minute is the mean, then it is exactly
C. 68%
in the middle of the data meaning 50% is above
D. 84%
and 50% is below, so the answer is B
E. 97.5%
Question 7
The percentage of 18-year-old students with pulse rates less than 53 beats/minute
or greater than 86 beats/minute is closest to
A. 2.5%
B. 5%
C. 16%
D. 18.5%
Lets look at this one a bit more closely
E. 21%
The pulse rates of a large group of 18-year-old students are approximately normally
distributed with a mean of 75 beats/minute and a standard deviation of 11
beats/minute.
34%
2.35%
34%
13.5%
13.5%
99.7%
2.35%
The pulse rates of a large group of 18-year-old students are approximately normally
distributed with a mean of 75 beats/minute and a standard deviation of 11
beats/minute.
This means that on our bell
shaped curve we can mark in
each of the values as shown
below
34%
2.35%
42
34%
13.5%
53
2.35%
13.5%
64
75
99.7%
86
97
108
We want to find the percentage of 18-year-old students with pulse rates less than 53
beats/minute or greater than 86 beats/minute is closest to
So we shade in the area that
this corresponds to...
34%
2.35%
42
34%
13.5%
53
2.35%
13.5%
64
75
99.7%
86
97
108
We want to find the percentage of 18-year-old students with pulse rates less than 53
beats/minute or greater than 86 beats/minute is closest to
So we shade in the area that
this corresponds to...
34%
2.35%
42
34%
13.5%
53
2.35%
13.5%
64
75
99.7%
86
97
108
As can be seen, the middle, unshaded region is the bit outside of this range and this
corresponds to 34% + 34% + 13.5% = 81.5%
100% - 81.5% = 18.5 %
34%
2.35%
42
34%
13.5%
53
2.35%
13.5%
64
75
99.7%
86
97
108
The pulse rates of a large group of 18-year-old students are approximately normally
distributed with a mean of 75 beats/minute and a standard deviation of 11
beats/minute.
Question 6
The percentage of 18-year-old students with pulse rates less than 75 beats/minute
is closest to
A. 32%
B. 50%
C. 68%
D. 84%
E. 97.5%
Question 7
The percentage of 18-year-old students with pulse rates less than 53 beats/minute
or greater than 86 beats/minute is closest to
A. 2.5%
B. 5%
C. 16%
D. 18.5%
E. 21%
So the answer is D
Correlation
• Scatterplots allow us to observe if there is a
pattern that exists between two variables. The
more the data resembles a straight line, the
more likely there is a linear relationship
between the two variables.
Moderate Positive Correlation
Strong Negative Correlation
Weak Positive Correlation
No Correlation
Moderate Negative Correlation
Pearson’s Product Moment Correlation
‘r’
• If r is close to -1, then there is a strong
negative linear correlation
• If r is equal to 0, then there is no linear
correlation
• If r is close to +1, then there is a strong
positive linear correlation
Linear Regression
• Linear Regression is all about fitting straight
lines to the given data.
• Be familiar with your equation for a straight
line
y=mx+c
• Know the difference between the
independent and dependent variables
Use the CAS Calculator
• If you don’t use the CAS Calculator in the end
of year exam, you are putting yourself at a
massive disadvantage compared with all those
other students in the state who are using it.
• Especially when dealing with data
calculations, the CAS Calculator will save you a
huge amount of time.
The 3-median method
A person’s weight is also known to be positively associated with their height. To
investigate this association for 12 men, a scatterplot is constructed as shown
below.
While there is a moderately strong positive linear relationship between weight
and height, there is a clear outlier. Because of this, it is decided to model the
relationship by fitting a 3-median line to the data displayed in the scatterplot.
Begin by splitting the data up
into 3 equal parts
4
4
1
3
3
1
2
2
2
1
3
4
Now number each of the
values in each section from left
to right
4
Find the middle point of each
of the points along the x-axis.
1
4
3
3
1
2
2
2
1
3
4
If there are an odd
number of points,
then one of the
points will be on
the line, if there
are an even
number, the line
will be in the
middle of the two
centre points
4
4
3
2
4
3
1
2
3
1
2
1
Now number each of the
values in each section from
bottom to top
4
4
3
2
4
3
1
2
3
1
2
1
Now find the middle point of
each of the points along the yaxis.
4
Mark each of the places where
the median lines cross
4
3
2
4
3
1
2
3
1
2
1
Rub out the un-important stuff
Draw a line through the two
outermost median points
Now if you are after the
gradient of the 3-median line,
you can work that out here as
the gradient of the line won’t
change again.
If you need to actually plot
the line however, there is
one more thing that you
need to do...
Measure the distance between
the middle median point and
the line and divide this by 3
To make things nice, lets say I
measure this and find that it is
6mm away from the line.
6mm ÷ 3 = 2mm
We then draw a new line that is
2mm closer to the middle
median point but that has the
same gradient.
This is the 3-median line
2mm
Least Squares Regression
I was talking to my housemate the other night about
the 3-median method (I know, sad isn’t it?) and he
was telling me that while it is a valid way of
interpreting data, he wouldn’t use it. ‘A least squares
regression’ he said ‘would make a much better
approximation for numerous reasons...’
He then went on to explain them,
but I’ll spare you that conversation
as even I found it rather boring.
Least Squares Regression
So how does the Least Squares Regression work?
Least Squares Regression lines work by finding the line that
results in the least distance between all the original data
points and the line being fitted. In this sense, it quite
literally could be considered the ‘closest fit’.
Residual Analysis
A residual analysis then looks at the vertical
distances between the regression line and the
data points to see if there are any patterns.
If there is a pattern, then it is likely that the
linear relationship that was supposed is in fact
not the true relationship between the two
variables.
Time Series
• Time Series are graphs that have time as the
independent variable
• There are 4 types of Time Series Trends
... and you have to know them
ALL!!!
Secular Trends
Is a trend that is steadily increasing of
decreasing
Seasonal Trends
A trend that fluctuates over a given period of
time (a season, can be summer, or footy
season, etc.)
Cyclic Trends
A trend that fluctuates upwards and
downwards but is not related to seasons
Random Trends
• Are trends that occur at random without any
particular pattern
Smoothing
• We can fit trend lines to time series data if it is
secular or random to see if there is a
relationship (i.e. if it increases or decreases
over time)
• In order to do this we need to smooth the
data to make it more linear.
Median Smoothing
Median smoothing takes the median of a
group of data points and uses these as a new
‘smoothed’ data set.
Moving Average Smoothing
With moving average smoothing, the original
time series data points are replaced with the
averages of the original points. This reduces
the large variation that might otherwise occur
in the data.
Deseasonalising Data
When dealing with Seasonal or Cyclic data, it
is difficult to see if there is a linear trend to
the data. By deseasonalising, we remove the
large variation from season to season and are
better able to see any trends that might occur.
Seasonal Indices
The seasonal indices tell you how the original
value compares with the seasonal average across
the whole year.
If the seasonal average is 1.4, it is 1.4 times the
yearly average, if it is 0.4, it is 0.4 times the yearly
average and so on.
Deseasonalised value = original value ÷ Seasonal Index
REMEMBER
When all the seasonal indices are added
together it should be equal to the number of
seasons that occur each year.
So if there are 4 seasons, the seasonal indices
should add up to 4.
If there are 10 seasons, the seasonal indices
should add up to 10.
Practice, Practice, Practice
I will run another revision study hall
where I cover the other modules from
the exams but until then, make sure you
do lots and lots of practice exams and
questions.
If you get stuck on a question, ask your
teacher, ask your classmates, ask
random people you meet on the street.
DON’T sit and wallow in your own self
pity as doing this won’t get you
anywhere.
The End