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B1andB2 · Jul 28, 2021

Original Poster said:

If i recall correctly all you have to do is use your calculator
change it to stat mode with frequency on then press the A+Bx function. You should see a table now
sub in the values of the table in the question into your calculator
press AC not on
then go like shift 1 "Reg" and press A and B to get those values
line has eqn y=Bx+ A i think, so just substitute and it should give you the answer i hope

B1andB2 · Jul 28, 2021

Hivaclibtibcharkwa said:
Aw shoot, my calculator can do this? Gosh daym.
Any way to do it with maths though?

yeah it can, wouldn't your teacher have shown it to you at school?

pre sure the other way is a very long, tedious and unnecessarily complicated process. It's probably shown in cambridge as a 'manual method' idk

jimmysmith560 · Jul 28, 2021

Back when I did statistics during my first year at uni, we were taught these:

In order to do this mathematically, I believe you have to go through these formulas (or if they're written differently in HSC textbooks). The values you need are as follows:

$\sum \:xy=304$
$n=8$
$\sum \:x^2=252$
$xbar=\frac{19}{4}$
$ybar=12$

What you need to do from there is plug the values into the formulas, giving you Bx (B1 hat) and A (B0 hat).

As B1andB2 mentioned, it's definitely a much longer process and very inefficient for a 1 mark question.

CM_Tutor · Jul 28, 2021

As far as I am aware, there are three options here:

Option 1: Skip the question and score no marks.

Option 2: Learn how to get the calculator to give you the equation.

Option 3: Learn how to do the problem manually.

Here is option 3:

$\begin{tabular}{ccccc}\text{Child}&\text{Exercise hours, $x_i$}&\text{TV hours, $y_i$}&\text{${x_i}^2$}&\text{$x_i y_i$} \\1&4&12&16&48 \\2&1&24&1&24 \\3&8&5&64&40 \\4&7&9&49&63 \\5&10&1&100&10 \\6&3&18&9&54 \\7&3&11&9&33 \\8&2&16&4&32 \\N=8&\text{$\sum x_i = 38$}&\text{$\sum y_i = 96$}&\text{$\sum {x_i}^2 = 252$}&\text{$\sum x_i y_i = 304$} \end{tabular}$

The slope of the required line is

$\begin{align*} m &= \cfrac{N\sum x_i y_i - \sum x_i \sum y_i}{N\sum {x_i}^2 - \left(\sum x_i\right)^2} \\ &= \cfrac{8 \times 304 - 38 \times 96}{8 \times 252 - 38^2} \\ &= \cfrac{-1216}{572} \\ &= -\cfrac{304}{143} \\ &= -2\cfrac{18}{143} \end{align*}$

and the $y$ -intercept is

$\begin{align*} b &= \cfrac{\sum y_i - m\sum x_i}{N} \\ &= \cfrac{96 - -\cfrac{304}{143} \times 38}{8} \\ &= \cfrac{96 \times 143 + 304 \times 38}{8 \times 143} \\ &= \cfrac{25280}{1144} \\ &= \cfrac{3160}{143} \\ &= 22\cfrac{14}{143} \end{align*}$

and thus, the least-squares regression line of best fit is

$y = -2\cfrac{18}{143}x + 22\cfrac{14}{65} = \cfrac{3160 - 304x}{143}$

and you can see that this matches the data. It has a negative slope. No exercise ( $x = 0$ ) corresponds to our $y$ -intercept at about 22.2 hours of TV, and no TV ( $y = 0$ ) goes with our $x$ -intercept at about 10.4 hours of exercise. By eye, this matches our plot.

Now, we can also be asked to find the correlation coefficient, $r$ , which is a measure of how well the line fits the data. It must satisfy $|r| \leqslant 1$ A good fit should have an $r$ near 1 for a positive correlation and near -1 for a negative correlation. A poor fit will have an $r$ near 0. It is usually reported as $r^2$ . In this case, we have a strong negative correlation as $r \approx -0.927$ and $r^2 = 0.859$ The formula is

$r = \cfrac{N\sum x_i y_i - \sum x_i \sum y_i}{\sqrt{n\sum {x_i}^2 - \left(\sum x_i\right)^2}\sqrt{n\sum {y_i}^2 - \left(\sum y_i\right)^2}}$

If you add a ${y_i}^2$ column to the table, you should find that $\sum {y_i}^2 = 1528$ , and hence can calculate that:

$\begin{align*} r &= \cfrac{N\sum x_i y_i - \sum x_i \sum y_i}{\sqrt{n\sum {x_i}^2 - \left(\sum x_i\right)^2}\sqrt{n\sum {y_i}^2 - \left(\sum y_i\right)^2}} \\ &= \cfrac{8 \times 304 - 38 \times 96}{\sqrt{8 \times 252 - 38^2}\sqrt{8 \times 1528 - 96^2}} \\ &= \cfrac{-1216}{\sqrt{572}\sqrt{3008}} \\ &= \cfrac{-76}{\sqrt{11 \times 13 \times 47}} \\ &= \cfrac{-76\sqrt{6721}}{6721} \\ &\approx -0.927036... \end{align*}$

Of course, the calculator could have spat out this result too, by the option 2 method.

Before anyone comments, it is true that there are different calculating formulae for $m$ , $b$ , and $r$ . They are all equivalent and will give identical results to mine above, assuming that I have made no mistakes. (ADDED BY EDIT: And @jimmysmith560 has posted one of them while I was typing all this up!

... Question to self... why did I type this up? lol END OF EDIT)

TAKE HOME MESSGAE: (And I am paraphrasing from the Marker's notes on quite a few 2020 trials)... It is important that ALL STUDENTS learn to use the Stats mode on their calculators for regression and correlation questions.

MY ADDENDUM: unless you'd rather learn the method to solve the problem manually.

CM_Tutor · Jul 28, 2021

jimmysmith560 said:
Back when I did statistics during my first year at uni, we were taught these:

View attachment 31283

In order to do this mathematically, I believe you have to go through these formulas (or if they're written differently in HSC textbooks). The values you need are as follows:

$\sum \:xy=304$
$n=8$
$\sum \:x^2=252$
$xbar=\frac{19}{4}$
$ybar=12$

What you need to do from there is plug the values into the formulas, giving you Bx (B1 hat) and A (B0 hat).

As B1andB2 mentioned, it's definitely a much longer process and very inefficient for a 1 mark question.

@Hivaclibtibcharkwa, note that @jimmysmith560 formulae and mine are equivalent. The values that he has given for

$\sum x_i y_i = 304 \quad \quad \text{and} \quad \quad \sum {x_i}^2 = 252 \quad \quad \text{and} \quad \quad N = 8$

match mine exactly, and the other two are equivalent as

$\begin{align*} \overline{x} &= \cfrac{1}{N}\sum x_i \qquad \text{by definition} \\ &= \cfrac{38}{8} \qquad \text{from my data} \\ &= \cfrac{19}{4} \qquad \text{matching jimmy's answer} \end{align*}$

and similarly for $\overline{y}$ and the different forms of the equation call for means (in jimmy's formulae) and sums (in mine).

CM_Tutor · Jul 28, 2021

Hivaclibtibcharkwa said:
So it seems like the question wants to be use a calculator. I’ve never seen this maths in the classroom before

Have you finished the syllabus yet? It is near the end of the year 12 Advanced course.

jimmysmith560 · Jul 28, 2021

Hivaclibtibcharkwa said:
What are you studying in uni?

The degree I'm studying is called Bachelor of Business (Advanced Business Leadership) and my major is Human Resource Management.

I had to do a statistics unit relevant to business in my first year. This was one of the concepts we covered (when I did Mathematics in year 12 statistics were not in the syllabus).

CM_Tutor · Jul 30, 2021

Hivaclibtibcharkwa said:
Well according to my teacher I have. But this is the first time I’m using my calculator to get an equation using a table

Well, this type of question appeared on the NESA Sample Paper (Q 37) plus several trial papers, including Blacktown Boys HS, Caringbah High, CSSA, Gosford High. Hurlstone Ag HS, North Sydney Girls High, Sydney Girls High, the Independent Schools paper, and the papers from companies that sell trials to multiple schools.

So, there are plenty of places you can look for practice... and this was a popular question last year on new content, and I would expect it on most trials and likely the HSC this year.

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