Hello and welcome to Mr Tompkins Ed Tech. In this video we will be taking another look
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. Those of you following
an IB Higher Level Mathematics or IB Biology course will also find the style of question
spookily familiar, so keep watching.
In the last video we looked at the single variable calculation function, and used it
to find mean and standard deviation.
This time we will be looking at how the Casio Graphical Calculator copes with two variable,
or bivariate data. We will be using the graphing tool to plot a scatter graph, and the 2 variable
CALC function to generate a value for the Pearson's Product Moment Correlation Coefficient,
or PMCC.
So here is the exam question. It concerns 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.
It is always important to read the question carefully as the context can give strong clues
as to the expected answer. In this case, we would expect to find that as the council increased
the charge, the number of vehicles entering the city would fall. This would imply a strongly
negative value for PMCC, and so if our calculation doesn't show us this, it should alert us that
we may have made a mistake somewhere along the line and we should probably go back and
check it. It's much better to find out you have made a mistake whilst you are in the
exam hall and can still do something about it, rather than when the results come in.
Ok, so lets start entering the data into the calculator and see what it gives us.
So first up, put the calculator in STATS mode, which is 2 on the main screen.
That brings up this screen, which has multiple columns to enter lists of data. This time
because we have bivariate data, we need two lists to hold our values from the data table.
Before we put the new data in, lets clear out anything that is already in the memory
from the last time we used it. First we press F6 to bring up the 2nd menu of functions,
followed by F4, which is the Delete All function, and F1 to confirm. This will clear out all
the values in the currently selected list. Now, move the cursor into the second column
and then press F4 followed by F1 again to get rid of List 2.
Right having now tidied up, we can start entering our new data.
We will use LIST 1 to hold the Charge, in £s, and LIST 2 to hold the average number
of vehicles per day.
So 4, 6, 8, 10, 12, 14, 16, 18, then tap the right arrow for list 2, and we enter 2.4,
2.5, 2.2, 2.3, 2.0, 1.8, 1.7, 1.5. Done.
Right, so F6 brings us back to the original menu of functions and we want to tap F1 to
graph the data.
The calculator lets you store up to 3 presets for graphs, and you can press the SET button
to define what you want these to be. At the moment I have graph 1 set up to draw a box
plot and graph 2 to draw a scatter graph and graph 3 to draw a pie chart, but you can set
these to whatever you want from the 20 or so available graph types the calculator can
draw.
Now for a scatter graph, you need to tell the calculator where to pull its x and y values
from. So we are using list 1 and list 2 for these. The frequency should be 1, and you
can choose what sort of mark type you want to show.
Once you have this set up, press EXIT, and select the preset you used. I used Graph 2,
so I'll press F2.
Unsurprisingly we have a scatter graph showing strong negative correlation. This fits with
the context, as we suspected that as the charge went up it would drive the number of vehicles
down.
Sadly we cant just print this out and stick it on our exam paper, but we can use it as
a visual check that our hand-drawn graph is correct. Once the graph is drawn, you can
use the trace function SHIFT F1 to show the actual plot points and their values and use
this to double-check the values on your drawn graph.
So having drawn your graph and checked it with your calculator, we are ready for the
main event - calculating PMCC.
Now PMCC is always a number between 1 and -1, with numbers close to 1 showing strong
positive correlation and close to -1 showing strong negative correlation. We have already
noted that our scatter graph clearly shows strong negative correlation, so we are expecting
an answer around negative 0.9 here.
Right lets hit the F2 CALC button, and quickly check SET to make sure we are configured correctly
for bivariate data. Check that the 2VAR XList points to List 1, and that the 2Var YList
points at List 2. If it doesn't for some reason, you can press the LIST button and set it correctly.
Now press EXIT, followed by F2 2VAR and see what numbers we get.
You can see it generates the totals, mean and standard deviation, along with the min
and max values for both the x and the y datasets.
We can use this screen to copy down the totals if they are not given in the question itself.
For the PMCC calculation, we will need n, the number of elements, Sigma X, Sigma Y,
Sigma X squared, Sigma Y squared, and Sigma XY. Lets write all those down.
n is 8, sigma x is 88, sigma x squared is 1136, scrolling down, Sigma y is 16.4, sigma
y squared is 34.52 and sigma xy is 168.6
Whilst your calculator will also churn out a value for PMCC, you will need to show method
to get all the marks. So lets write down the PMCC formulae.
We would normally need to evaluate these 3 formula and then sub our answers into the
4th to find PMCC, but as we are smart and we are going to use the calculator to take
the sting out of this a little, to minimise our effort and so allow us more time for other
more challenging parts of the exam paper.
We are going to do all 4 at once. This way, we don't actually need to evaluate each section
separately, thus minimising the risk of making a calculation error, and can just write down
the final answer.
So here goes. It's a bit of a beast, but with a little care it's not too bad.
Now, there is no need to actually type this behemoth into the calculator. We just need
to go to EXIT and choose F3 for the regression function. We will cover the regression function
in much more detail in the next video. For now we will just select F1 for X, or linear
regression. Now both the y-on-x or x-on-y regression functions will give us the same
value of PMCC, so it doesn't matter which one I choose. So lets go for ax+b.
PMCC is given by this r value, so I just need to copy that down to an appropriate degree
of accuracy. So minus 0.960 to 3 significant figures. Done. Easy peasy, and our answer
is reassuringly close to -1, as we expected it to be.
Now the mark scheme states 2 method marks and 1 accuracy mark for this question. You
get the first method mark for a correct substitution into any S formula, and the second for a correct
substitution into the r formula. We have done both at the same time. We will get the third
and final accuracy mark by writing down the correct value for PMCC.
Now there is no getting around the fact that you need to learn these formula, and you will
also need to be able to substitute carefully into them to gain the method marks. However,
the graphical calculator has done the hard number crunching, so you don't have to, saving
you time and minimising the risk of making a typing error in this otherwise long and
intricate calculation.
So that's all for this video. You should now know how to use your Casio graphical calculator
to plot a scatter graph and to find PMCC the easy way.
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, I'll be delving into the linear regression function
in much more detail. I'll be showing you how to find y-on-x and x-on-y regression lines,
along with some explanation on when you should use one or the other. So please hit subscribe
so you will be notified of this and any future uploads.
Thanks for watching and hope see you next time.
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