VCE General & Further Maths Question Thread!

[keltingmeith]

Clarification:
You can quantify skew - there are formulas for you to figure out if something is symmetric and, if not symmetric, how skewed (say, +2, -3 or even +6) it is. This is definitely not in Further, you only need to identify if something is negatively skewed, symmetrical or positivity skewed.

Tl;dr, something can be slightly positively skewed, but you don't need to know this for Further.

[Garden]

Thanks Euler!

[Vexna]

Can someone explain the whole process of finding seasonal indices, deseasonalising the data and adjusting it?

This is somewhat confusing and not my strong point when it comes to maths.

[scarletmoon]

Ok so when I do practise exams for further,I do them with my CAS in radian mode (I always forget to change it to degrees bc I also do methods) and I get the right answers. Today I realised that I did my sac with my CAS in radian mode.

Would my IQR, regression equation, Pearson coefficient be different, should I go back next lesson and redo the calculations with my CAS in degree mode?

[keltingmeith]

Ok so when I do practise exams for further,I do them with my CAS in radian mode (I always forget to change it to degrees bc I also do methods) and I get the right answers. Today I realised that I did my sac with my CAS in radian mode.

Would my IQR, regression equation, Pearson coefficient be different, should I go back next lesson and redo the calculations with my CAS in degree mode?

No, they should all be the same. Degrees vs Radians is only an issue for trig stuff.

[MightyBeh]

Can someone explain the whole process of finding seasonal indices, deseasonalising the data and adjusting it?

I couldn't think of a good way to explain in-depth, so here's a general thing with a bunch of examples. Everything is to 4dp for convenience. If you were doing this in a SAC or exam, I would recommend you do all these calculations on your calculator (If you didn't know, there's a bunch of neat functions in the L&S application, if you've ever worked with Excel formulas they're pretty much the same).

Seasonal indices tell us how a particular season (Generally a day, month or quarter(3 months). Otherwise I believe it's defined as cyclical) compares to the average season.

The formula for a season index is:

It's worth noting that the sum of the seasonal indices equals the number of seasons. (For example, a has a SI of 0.75, b has 1.75, c has 1.0 and d has 0.50.

All that is really kind of abstract though, so it's easiest to explain (and hopefully understand) with an example.

So here's our data:

Step 1 - Seasonal Average

So, first we work out the 'seasonal average'. This is done by adding up all of a seasons's values and dividing them by the number of seasons. In our data set, the seasons are quarters.

working out the seasonal average

The formula for the seasonal average is:

Here's that in context with our data.

Step 2 - Finding the Seasonal Index

So here, we have another table (With bigger gaps! What an advance in technology) :

There are kind of two parts to this step; first we have to find the seasonal index for each quarter per year (We'll call this section 2a, because everyone loves nested spoilers) and then we have the work out the "average" seasonal index for each quarter, overall (This is 2b. feel free to skip ahead).

2a.

This is a pretty easy step, it just takes some time and (possibly, depends on your preferences) fiddling with your CAS. What you need to do is divide each value by its corresponding seasonal (yearly) average, like so:

For for the first quarter in 2003:


We're rounding to 4dp for simplicity here, but I'd definitely recommend using exact values until you have to give a decimal or percentage answer because they're much more accurate and you're a lot less likely to make a rounding error... if you haven't done any rounding?

Here's our table with our approximate values in it; now we're at 2b.

2b.

Here it is in terms of the first quarter:

Where from our original formula, Qn becomes Q1 and Nyears becomes 3 (years). I'm going to show Q2 as well, just to be sure:

Here's that on our table:

Step 3 - Deseasonalising

*If you're doing this with your calculator, I'd recommend you set your SI to a variable so you're less likely to make an input error. In a calculator page you can do this by entering the following:
**I'd also recommend doing this in a document as opposed to Scratchpad if you don't already because you're less likely to forget about it and mess up some other calculation for some other work. As a general rule I'd recommend making a new document for each class or (long) question dealing with data.

Code: [Select]

a:=0.7233, b:=1.2294, etc.To deseasonalise data, you use the following formula:

So if we were to do this for the first quarter of 2003, it would be:

if you're doing this on paper, it depends on the grouping on your graph, but usually rounding to 2dp or nearest whole number is the way to go.
Same thing for the second quarter of 2003:

Keep going until you end up with this:

(Rounded to the nearest whole number, again for convenience. You can keep it exact / 4dp if you're on your calculator)

Step 3a - Graphing the Deseasonalised Data

Same table as before, but here instead of using 20031,20032,20033... 20054 for our x axis (Always time with time series), we convert the years/quarters/seasons/etc. to an easier to read (and graph) value, like so:

I've put them into ordered pairs here so that it's clear how to graph them. I don't actually have a copy of the graph here but if it's really unclear (which it probably is, I'm not amazing at English), I can draw one for you. Just let me know

Step 4 - Predicting Using the Regression Line

Workout the regression line your preferred way with the data, it's not a huge deal. I generally find it's quickest and easiest to just get it when I plot it on the graph page, but that's up to you. Hopefully you should get something like this:

But it's also pretty common to have to write it in terms of the variables, so:
[text]Sales=5.6970+01364*QuarterNumber[/tex]

Now, you're nearly 100% guaranteed to be asked something along the lines of "Calculate the deseasonalised value for..." or "Make a prediction for the..." after that, so we're going to deal with both of those situations.

1) Using the equation of your regression line, calculate the deseasonalised value for the second quarter of 2006.
So after reading the question, (hopefully) you've noticed that the second quarter of 2006 doesn't actually exist. We'll first have to work out the Time(t) value (The "Quarter Number in our equation above"). To do that you could either:
a) Count the quarters until you're at the second quarter of 2006
b) Use some simple algebra to find it:

Where 3 is the number of full years preceding 2006, 4 is the number of quarter per year and "+2" is to calculate the second quarter.


So now that we know that t=14, we can substitute that into our equation and solve for the deseasonalised value.
It's fastest to use your calculator like so:

Code: [Select]

Solve(y=5.6970+(0.1364*14),yand you should get:

(round to the nearest whole number because we're talking about diiscrete data, specifically sales.)

2) "Use your deseasonalised value to find the correct seasonalised sale", or "predict the number of sales made in the second quarter in 2006".
With these, the second one especially, it's important to read the question (and highlight, if you so choose) that you're being asked for the seasonalised value, not the deseasonalised one. Lots of people forget to convert it.

We convert it by:

In English: Multiply the deseasonalised value by the appropriate seasonal index to find the sales value.

So, in terms of our question:

I think that's everything to do with working with seasonal indices, but I mostly copied this from my notes document and fixed some things so that they'd work on the forum. What makes sense to me might not make sense to you, so feel free to make me explain stuff if you need it. I'd also like to mention that I haven't proof read this so you can also yell at me for really terrible formatting mistakes. Don't worry, I'll only feel bad for not noticing them myself <3
^{sorry I'm using so many images, it's a lot neater than writing out every equation in LaTeX, and while I could use a BBCode table, these are screenshots from a summary sheet I made so it's a lot easier to share.}

[RazzMeTazz]

Would the terms 'bar graph' and 'column graph' be interchangeable, or if the question asked for  bar graph would you specifically have to draw bars that are going horizontally, and if it asked for column graphs would you have to draw vertical bars?

Because, looking through different textbooks and other resources, the two terms seem to be used synonymously?

Thanks

[Escobar]

if this came up in the applications SAC (excel), how would I answer it?
using mean, range, iqr etc,describe your data in general terms

does this mean just stating the mean, range etc lol
what does 'in general terms' mean?

[MightyBeh]

Would the terms 'bar graph' and 'column graph' be interchangeable, or if the question asked for  bar graph would you specifically have to draw bars that are going horizontally, and if it asked for column graphs would you have to draw vertical bars?

I'm not actually 100% on this one, but I'm pretty sure that they're used synonymously. I checked MathIsFun, and that seems to say they are the same; but sources based around MS Excel seem to be saying that bars are horizontal and columns are vertical (which is how Excel organises them so it's user friendly, I believe). If you're doing a question designed for an Excel based SAC, I'd differentiate between 'bar' and 'column'. Otherwise you should be safe just drawing it vertically (and I don't think I've seen a test question ask for anything other than that before - maybe a textbook though).

If you're really worried about it, judge it by the marks (say, 4 marks for drawing a bar chat showing all important features (pretty unlikely mark distribution, but you know)) or ask your teacher if they'll be marking on it.

Unless you're talking in terms of exams; I'd like to think they'd tell you if they wanted something specific. Then again, it's VCAA. They'll find a way to make it harder to understand.

if this came up in the applications SAC (excel), how would I answer it?
using mean, range, iqr etc,describe your data in general terms

does this mean just stating the mean, range etc lol
what does 'in general terms' mean?

Not sure on what 'general terms' means. It could mean not quoting the statistics in your comment, or rounding to integers or something like that. As a general rule, more information is better than less (as long as it's concise).

Since the question includes 'etc.' (I assume?) I'd say it's just asking you to write a report on it. What statistics you use kind of depends on the previous question(s); if you were asked to make a box plot I'd use the median(centre), IQR (spread), and comment on the shape and if there were any outliers. Most importantly all you need to do is cover all your bases - Shape, outliers, centre and spread (SOCS, if it helps you remember. It's always scribbled somewhere on my work).

I think it might just be an awkwardly phrased question though. As always, feel free to correct me if I'm wrong

[Garden]

Hi, I've got a few questions from broad to fairly specific, any help is greatly appreciated.

Is there an assumption that by using Pearson's correlation coefficient and r being lower than -0.24 or higher than 0.24, that the relationship is linear (assuming that the relationship is numeric and without outliers). Or can r be lower than -0.24 and higher than 0.24 and not be linear, even thought a weak/moderate/strong positive/negative relationship exists between the two variables.

For example, r in this case is 0.3515 (to 4 decimal places). From this, I've derived that a weak positive relationship exists between the two variables. Should I state this relationship is a weak linear positive relationship?



I've also included the scatterplot if this' of any assistance.



After further analysis using a residual plot, I've concluded that there is a clear pattern which confirms that the use of a linear equation to describe the relationship between the variables in non-suitable. Is this correct?

Can I jump from saying that a linear relationship exists between the variables when discussing r and then build a contradicting conclusion based on a residual analysis?



Thanks again. 

[scarletmoon]

Can deseasonalised values be larger than the actual value??

[StupidProdigy]

Can deseasonalised values be larger than the actual value??

Yes. They can be either bigger or smaller. Think about if you divide your actual value by the seasonal index (to get the deseasonalised figure obviously) and what happens when the seasonal index is <1.00 or great than >1.00. So the opposite applies for getting the actual value.
e.g deseasonalised figure is 3234 dogs, seasonal index is 1.4 in winter and 0.7 in summer.
calculating the actual value for winter dogs is given by 3234x1.4=4528 dogs. but if we want the summer dogs actual value then 3234x0.7=2264.
Make sense? I just said dogs because I couldn't think of anything

[MightyBeh]

For example, r in this case is 0.3515 (to 4 decimal places). From this, I've derived that a weak positive relationship exists between the two variables. Should I state this relationship is a weak linear positive relationship?

After further analysis using a residual plot, I've concluded that there is a clear pattern which confirms that the use of a linear equation to describe the relationship between the variables in non-suitable. Is this correct?

Can I jump from saying that a linear relationship exists between the variables when discussing r and then build a contradicting conclusion based on a residual analysis?

Thanks again. 

I'd say yes, call it linear if you're using r. When you use Pearson's, you're making the assumptions that: (Quoth Essentials, explanations are great. Love 'em)

• Variables are numeric
• The relationship is linear
• and that there are no outliers. The correlation coefficient can give a misleading indication of the strength of a linear relationship if there are outliers present*

You can always transform the data later on and state that it (Or, its r² value) is the more appropriate because... or that it implies/shows that the natural data is not linear because... (so yes, you should be allowed to change your mind)

Seems pretty clear to me that your data is non-linear, but it probably still has quite a low Correlation/Determination.

*Although I haven't actually seen this one in practice. Maybe I just haven't run into a problem with outliers. I don't think it applies to your data anyway, so you should be fine.

[Garden]

I'd say yes, call it linear if you're using r. When you use Pearson's, you're making the assumptions that: (Quoth Essentials, explanations are great. Love 'em)

• Variables are numeric
• The relationship is linear
• and that there are no outliers. The correlation coefficient can give a misleading indication of the strength of a linear relationship if there are outliers present*

You can always transform the data later on and state that it (Or, its r² value) is the more appropriate because... or that it implies/shows that the natural data is not linear because... (so yes, you should be allowed to change your mind)

Seems pretty clear to me that your data is non-linear, but it probably still has quite a low Correlation/Determination.

*Although I haven't actually seen this one in practice. Maybe I just haven't run into a problem with outliers. I don't think it applies to your data anyway, so you should be fine.

Cheers Mighty!

[n.a]

Hi!
Just wondering if someone could help me with this question from the 2014 FM exam:

The seasonal index for heaters in winter is 1.25.
To correct for seasonality, the actual heater sales in winter should be

A. reduced by 20%
B. increased by 20%
C. reduced by 25%
D. increased by 25%
E. reduced by 75%

Have a SAC on Tuesday! 
Thanks!!