Hello and welcome to Mr Tompkins Ed Tech. This is the third video in a series looking
at the Casio Graphical Calculator and how we can use it to solve exam style problems
taken from real A level S1 Statistics and probability papers. Students following IB
mathematics or Biology courses are welcome to follow along too, as the style of questions
will be very familiar.
In previous videos we have looked at finding mean and standard deviation, and Pearson's
PMCC. This time I'll be showing you how to find least squares regression lines, and how
to decide between y-on-x or x-on-y. We'll also be looking at using regression lines
to make estimates and discussing the reliability of these.
Least squares regression is essentially a technique for finding the equation of a line
of best fit. In GCSE we just had to put our ruler on a scatter graph, wiggle it around
a bit until it looked about right and then draw this on . Not very mathematical really.
Least squares regression imagines drawing squares from the plotted points to a line
of best fit angled through the "Centroid" or x bar, y bar coordinate, and then mathematically
jiggling this about until the areas of all the squares are minimised. This graph shows
y-on-x regression. You can tell because the squares are formed by dropping lines vertically
from the plotted points to the regression line.
You can see as I angle it, those squares get larger and smaller
and we are aiming to get them all as small as possible so that when we add them up the area is minimised.
In x-on-y regression lines, the squares are formed slightly differently, by drawing horizontal
lines from the the plotted points to the regression line, but again the process aims to minimize
the sum of these areas. These two types of squares give slightly different values, resulting
in two slightly different lines of best fit. You will need to know when it is appropriate
to use one or the other of these lines, and examiners will often probe your understanding
of this process in the exam.
So lets get started. This is a continuation of the question in the last video, concerning
itself with a city council's attempt to reduce traffic congestion by charging motorists to
enter the city centre. The council started with a charge of £4 in the first year, and
increased it by £2 in each subsequent year.
We already entered the data into the calculator in the last video. If you are not sure how
to do this, then pause this video and go back and watch the one on PMCC first. There is
a link in the description below, or you can tap on the card on the screen.
Now, before we calculate anything, we first need to decide on the "appropriate" regression
line to use, which means choosing between either y-on-x or x-on-y.
To do this we have up to three checks to make:
1. Check the context 2. Check the scale
3. Check the lookup value
Firstly, check the context. Which value is "controlled", or set, and which is "dependant",
or measured?
Now in this question the council has direct control over the amount Charged to enter the
city. It cannot directly control how many vehicles will turn up in the city on any given
day, but they can certainly measure this with traffic cameras and sensors. So the charge,
or x value, is controlled variable and the number of vehicles, or y value, is dependent variable.
This tells us we need to use the y-on-x regression line in this question. The context is clear,
and there is no need to carry out the 2 further checks.
If in another question, it isn't clear from the context which variable is controlled and
which is dependent, you would next check the scale. Does one of them consist of nice whole
numbers which are ordered and incrementing steadily,
and is there one with lots of decimals, jumping about the number line?
Here the Charges are all whole values and going up in 2s. It is clearly controlled.
The number of vehicles are unordered decimals
so these are the measured dependent values.
If and only if you cannot discern the controlled and dependent variable from either the context
or the data, then there is one final check to make.
3. Check the lookup value.
In this question we are being asked to find the y value or number of vehicles when the
x value, or charge is £20.
So in the absence of any other clues, if you are finding y from x, you should use y-on-x.
Now all 3 of our checks tell us to use y-on-x here. But remember these checks are in order
of importance, so context is most important, and over-rules all other considerations.
So we want the y-on-x regression line, which is given by the formula y = a + bx.
We will need to find b, the gradient of the regression line first. Now the calculator
generates this value for us, but we need to get all 5 of those marks, so will need to
show where it comes from. The calculation for b uses the same Sxx and Sxy formula we
used in the last video for finding PMCC. The calculator doesn't generate these directly
unfortunately, but it does give us all the totals required. If they are not given in
the question, you can find them by pressing F2 for the Calc function, followed by F2 for
2 variable calculations. So copying them onto our paper, we can now substitute them directly
into the formula for b.
As with PMCC, we wont actually type this expression into the calculator. There is no point as
it already knows the value of b.
On the calculator, we can find this by pressing the F2 Calc function, F3 for Regression,
F1 for linear regression, followed by F2 for y-on-x regression. (F1 gives us the other
sort of regression - the x-on-y regression line).
Now you can see we have the values displayed for a and b, along with r which we already
know is PMCC.
Let's write down the value for b, and also our method for finding a.
We can now write down the full equation for the y-on-x regression line given by y=a +bx.
Now we have the equation of the regression line, exam questions will often go on to ask
us to use it to estimate certain values.
Here we are asked to find y when x is 20. Simply sub this into our equation to find y,
Don't forget to check the units carefully.
Here y is measured in millions, so the correct answer is 1.42 million vehicles
The last part of the question asks about reliability. There are two factors affecting reliability
and you will need to reflect on these and make a decision. Firstly strength of correlation.
For the estimate to be reliable there must be high correlation,
so anything above 0.9 for positive correlation,
or below -0.9 for negative correlation. In this question we have negative correlation
and calculated PMCC to by -0.96 so we've clearly got high correlation here.
The second consideration is whether the value we are looking for is within the given range
of data or outside it. If it is within, we are interpolating and the estimate will be
reliable. If the value we are looking up is outside the given range, then we are extrapolating
and the estimate is unreliable.
Now here 20 is just outside the data given so technically we are extrapolating, indicating
the estimate might be unreliable. But because correlation is so high and because it is only
just outside the range, the correct conclusion would be that it IS reliable.
So I'm going to write "correlation is high (-0.96) and is just outside the given data,
so the estimate is reliable".
Now lets review the marks for our work.
We have chosen the correct regression line, earning our first mark.
We gain a method mark for the substitution,
and accuracy mark for finding b.
a second method mark for our calculation for a,
and second accuracy mark for the correct equation.
For the second part, a method mark for subbing 20 into our equation, and an accuracy mark
for the correct answer.
The final 2 marks are B marks, which means they are independent of method or awarded
for correct statements. We earn 1 for high correlation and 1 for just outside range and
our conclusion. Note that often mark schemes will award you the marks either way as long
as you give a valid reason, so here saying no because it's extrapolating would still
earn you the marks.
So that's all for this video. You should now know how to choose the correct least squares
regression line, and to use your Casio graphical calculator to find it's equation. You also
now know how to decide whether estimates found will be reliable or unreliable.
Please like if you found it useful, and leave a comment if you have any questions or suggestions.
In the next video in this series, we'll be taking a look at discrete random variables
and how we can use the Casio graphical calculator to find probability ranges and thus avoid
having to use those strange throwback look-up tables. So please hit subscribe so you will
be notified of this and any future uploads.
Thanks for watching and hope see you next time.