Find regression equation and correlation coefficient,

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Newcastle University Mathematics and Statistics

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

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Name	Status	Author	Last Modified
Find regression equation and correlation coefficient,	draft	Newcastle University Mathematics and Statistics	20/11/2019 14:50
Linear regression and correlation - pub sales	Ready to use	Mario Orsi	16/02/2021 17:59

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Question statement

{owner} owns the {pub}. {owner} believes that sales of {beverage} in the pub are linked to the average monthly temperature, with higher sales being recorded in months with higher temperatures. To investigate, {owner} records the average monthly temperature in the local town over a period of one year ( $X$ degrees Celsius), along with total monthly sales of {beverage} ( $Y$ hundred pounds). The results are shown in the table below:

Month	$\var{obj[0]}$	$\var{obj[1]}$	$\var{obj[2]}$	$\var{obj[3]}$	$\var{obj[4]}$	$\var{obj[5]}$	$\var{obj[6]}$	$\var{obj[7]}$	$\var{obj[8]}$	$\var{obj[9]}$	$\var{obj[10]}$	$\var{obj[11]}$
$X$ (temperature)	$\var{r1[0]}$	$\var{r1[1]}$	$\var{r1[2]}$	$\var{r1[3]}$	$\var{r1[4]}$	$\var{r1[5]}$	$\var{r1[6]}$	$\var{r1[7]}$	$\var{r1[8]}$	$\var{r1[9]}$	$\var{r1[10]}$	$\var{r1[11]}$
$Y$ (sales, £100s)	$\var{r2[0]}$	$\var{r2[1]}$	$\var{r2[2]}$	$\var{r2[3]}$	$\var{r2[4]}$	$\var{r2[5]}$	$\var{r2[6]}$	$\var{r2[7]}$	$\var{r2[8]}$	$\var{r2[9]}$	$\var{r2[10]}$	$\var{r2[11]}$

You are given the following information:

$X$	$\sum x=\;\var{t[0]}$	$\sum x^2=\;\var{ssq[0]}$
$Y$	$\sum y=\;\var{t[1]}$	$\sum y^2=\;\var{ssq[1]}$

Also you are given $\sum xy = \var{sxy}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

ch	integer	7
prediction	integer	23
b1	number	0.45
owner	string	Peter
sxy	number	3439
res	list	List of 12 items
spxy	number	347.25
ls	number	24.34
tol	number	0.001
tcorr	number	0.6399742318
a	number	13.980
ssq	list	[ 2487, 5629 ]
sumr	number	272.887
pub	string	Cross Keys Pub
a1	integer	14
thisval	integer	17
corr	number	0.64
tsqovern	list	[ 1850.0833333333, 5166.75 ]
b	number	0.545
obj	list	List of 12 items
r1	list	List of 12 items
r2	list	List of 12 items
ss	list	[ 636.9166666667, 462.25 ]
tol1	number	0.01
n	integer	12
beverage	string	cold-filtered lager
t	list	[ 149, 249 ]
sc	integer	19
rsquared	number	0.410

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xxxxxxxxxx
 
random(0..11)

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Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

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Parts

Part a) Gap-fill
1. Gap Number entry
  Add
Part b) Gap-fill
1. Gap Gap 0. Number entry
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2. Gap Gap 1. Number entry
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1. Step Information only
Part c) Gap-fill
1. Gap Number entry
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Calculate the sample correlation coefficient $r$ for these data:

$r=\;$Unnamed gap (enter to 2 decimal places).

Advice

For part a) you calculate $r$ using:

\[r=\frac{S_{XY}}{\sqrt{S_{XX} \times S_{YY}}}\] where :

$\displaystyle S_{XY}=\sum xy - n\overline{x}\overline{y}$

$\displaystyle S_{XX}=\sum x^2 - n\overline{x}^2$

$\displaystyle S_{YY}=\sum y^2 - n\overline{y}^2$

For part b): The regression line has equation:

$\simplify[all,!collectNumbers]{Y={a}+{b}X}$ and this is displayed below:

{regfun(r1,r2,max(r1)+10,max(r2)+10,rsquared,sumr)}

For part c):

Predicted sales whem $X=\var{thisval}^{\small o}$C:

\[\begin{align} Y&=\simplify[all,!collectNumbers]{{a}+{b}* {thisval}}\\
&=\var{{a+b*thisval}}\\
&=\var{prediction}
\end{align}\] to nearest whole number of hundreds of pounds.

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Part	Test	Passed?
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Information only
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Find regression equation and correlation coefficient,

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Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Find regression equation and correlation coefficient,

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Newcastle University Mathematics and Statistics 9 years, 2 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`