Calculate moving averages, perform linear regression and give seasonal adjustments,

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Norah Storey

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Newcastle University Mathematics and Statistics 9 years, 3 months ago

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Question statement

{This} has recently opened for business and is open 5 days a week, Monday, Tuesday, Thursday, Friday and Saturday. The number of orders each day during the first three weeks of business is shown below:

{table([["7/4/08",{m1},{t1},{th1},{f1},{s1}],["14/4/08",{m2},{t2},{th2},{f2},{s2}],["21/4/08",{m3},{t3},{th3},{f3},{s3}]],[" ","Mon","Tues","Thur","Fri","Sat"])}

Give any introductory information the student needs.

			Name	Type	Generated Value

f1	integer	11
f2	integer	12
f3	integer	13
sdf2	decimal	dec("2.4")
sdf1	decimal	dec("2.8")
thisdate	string	Thursday, 1st May
t2	integer	6
saf	decimal	dec("2.6")
mm3	rational	51/5
mm2	rational	44/5
this	string	A plasma TV shop
mt2	rational	9
sas	decimal	dec("4.9")
t1	integer	5
sat	decimal	dec("-3.2")
asth	decimal	dec("0.68")
tms	decimal	dec("-0.06")
mth3	rational	53/5
sam	decimal	dec("-5")
mth1	rational	8
sdth3	decimal	dec("1.4")
s3	integer	16
s2	integer	15
s1	integer	13
sds1	decimal	dec("4.6")
th3	integer	12
th2	integer	10
th1	integer	8
m1	integer	3
m3	integer	5
tom	decimal	dec("5.454545454545454545454545454545454545455e-2")
sath	decimal	dec("0.67")
sdm3	decimal	dec("-5.2")
adj	decimal	dec("0.68")
thisday	string	Thursday
sdt3	decimal	dec("-3.4")
sdt2	decimal	dec("-3.0")
ass	decimal	dec("4.91")
comm1	string
tol	number	0.01
tol1	integer	1
ast	decimal	dec("-3.20")
m2	integer	4
meany	rational	512/55
testal	decimal	dec("7.149090909090909090909090909090909090909")
t3	integer	7
sdm2	decimal	dec("-4.8")
mf2	rational	48/5
comm	string
asf	decimal	dec("2.61")
sumysquared	rational	24034/25
sdth1	decimal	dec("0.0")
syy	rational	446/55
sm	decimal	dec("-5.06")
asm	decimal	dec("-5.00")
mth2	rational	47/5
testbe	rational	149/550
om	decimal	dec("0.055")
sdth2	decimal	dec("0.6")
summa	rational	512/5
estbe	decimal	dec("0.27")
mf1	rational	41/5
meant	integer	8
sumty	rational	849
sth	decimal	dec("0.62")
ms2	rational	49/5
ms1	rational	42/5
st	decimal	dec("-3.26")
ss	decimal	dec("4.85")
ms	decimal	dec("-0.06")
mt3	rational	52/5
ti	integer	18
sds2	decimal	dec("5.2")
sf	decimal	dec("2.55")
estal	decimal	dec("7.15")
estord	decimal	dec("13")

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
th1+random(1..3)

JME function reference

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`

← Depends on:

→ Used by:

f2
mf1
mm2
ms1
mth1
s1
sdf1

This variable doesn't seem to be used anywhere.

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

Complete the following table, showing the moving averages for the number of orders based on a five-observation cycle. The first one has been done for you. Input all moving averages to one decimal place.

	Mon	Tues	Thur	Fri	Sat
7/4/08	*	*	$\var{mth1}$	Gap 0.	Gap 1.
14/4/08	Gap 2.	Gap 3.	Gap 4.	Gap 5.	Gap 6.
21/4/08	Gap 7.	Gap 8.	Gap 9.	*	*

Advice

The completed moving average table is as follows:

	Mon	Tues	Thur	Fri	Sat
7/4/08	*	*	$\var{mth1}$	$\var{mf1}$	$\var{ms1}$
14/4/08	$\var{mm2}$	$\var{mt2}$	$\var{mth2}$	$\var{mf2}$	$\var{ms2}$
21/4/08	$\var{mm3}$	$\var{mt3}$	$\var{mth3}$	*	*

The mean value of $T$ is $\overline{t}=\frac{3+4+\cdots+13}{11}=\frac{88}{11}=8$ .

The mean value of $Y$ is $\overline{y}=\frac{\var{summa}}{11}=\var{meany}$ .

We have:

$\begin{align}S_{TY}&=\sum ty-11\overline{t}\overline{y}=\var{sumty}-11\times 8 \times\var{ meany}\\ S_{YY}&=\sum y^2-11\overline{y}^2=\var{sumysquared}-11\var{meany}^2=\var{syy}\\ S_{TT}&=\sum t^2-11\overline{t}^2=814-11\times 64=110 \end{align}$

Estimate for $\beta$ is $\frac{S_{TY}}{S_{TT}}=\var{testbe}=\var{estbe}$ to 2 decimal places.

Estimate for $\alpha$ is $\overline{y}-\beta\overline{t}=\var{meany}-\var{estbe}\times\var{meant}=\var{testal}=\var{estal}$ to 2 decimal places.

So the linear regression equation is $Y=\simplify{{estal}+{estbe}T}+\epsilon$ coefficients to 2 decimal places.

The following table shows the calculated seasonal deviations and seasonal means for each day of the week. These are obtained by taking the moving average data away from the original data on orders. The seasonal means for the days are obtained by taking the means in each column.

	Mon	Tues	Thur	Fri	Sat
7/4/08	*	*	$\var{sdth1}$	$\var{sdf1}$	$\var{sds1}$
14/4/08	$\var{sdm2}$	$\var{sdt2}$	$\var{sdth2}$	$\var{sdf2}$	$\var{sds2}$
21/4/08	$\var{sdm3}$	$\var{sdt3}$	$\var{sdth3}$	*	*
Seasonal Mean	$\var{sam}$	$\var{sat}$	$\var{sath}$	$\var{saf}$	$\var{sas}$

d) The seasonal effects for each day of the week are calculated by first finding the means of all the seasonal deviations found in the last table (not including the seasonal means in the last row). Then you take this away from the seasonal mean for each day.

We find that the mean of the seasonal deviations is:

$\simplify[all,!collectNumbers]{({sdth1} + {sdf1} + {sds1} + {sdm2} + {sdt2} + {sdth2} + {sdf2} + {sds2} + {sdm3} + {sdt3} + {sdth3}) / 11} = \var{om}$ to 3 decimal places.

{comm}

Seasonal effect for Monday: $S_M=\;\simplify[all,!collectNumbers]{{sam}-{om}}=\var{sm}$ to 2 decimal places.

Seasonal effect for Tuesday: $S_T=\simplify[all,!collectNumbers]{{sat}-{om}}=\var{st}$ to 2 decimal places.

Seasonal effect for Thursday: $S_{Thu}=\simplify[all,!collectNumbers]{{sath}-{om}}=\var{sth}$ to 2 decimal places.

Seasonal effect for Friday: $S_F=\simplify[all,!collectNumbers]{{saf}-{om}}=\var{sf}$ to 2 decimal places.

Seasonal effect for Saturday: $S_{Sa}=\simplify[all,!collectNumbers]{{sas}-{om}}=\var{ss}$ to 2 decimal places.

We further adjust the seasonal effects for each day by finding the mean of the seasonal effects we have just found and then taking this away from each of the seasonal effects.

We find that the mean of the seasonal deviations is:

$\simplify[all,!collectNumbers]{({sm} + {st} + {sth} + {sf} + {ss} ) / 5} = \var{ms}$ to 3 decimal places.

{comm1}

Adjusted seasonal effect for Monday $\;=\simplify[all,!collectNumbers]{{sm}-{ms}}=\var{asm}$ to 2 decimal places.

Adjusted seasonal effect for Tuesday $\;=\simplify[all,!collectNumbers]{{st}-{ms}}=\var{ast}$ to 2 decimal places.

Adjusted seasonal effect for Thursday $\;=\simplify[all,!collectNumbers]{{sth}-{ms}}=\var{asth}$ to 2 decimal places.

Adjusted seasonal effect for Friday $\;=\simplify[all,!collectNumbers]{{sf}-{ms}}=\var{asf}$ to 2 decimal places.

Adjusted seasonal effect for Saturday $\;=\simplify[all,!collectNumbers]{{ss}-{ms}}=\var{ass}$ to 2 decimal places.

We use the regression equation found above $Y=\simplify[all,!collectNumbers]{{estal}+{estbe}T}+\epsilon$ to estimate the number of orders on {thisdate}.

But we have to adjust for seasonality using the adjusted seasonal effect found above by adding on the adjusted seasonal effect for {thisday} i.e. $\var{adj}$ .

Note that $T=1$ corresponds to 7/4/08 and hence $T=\var{ti}$ for {thisdate}.

So putting $T=\var{ti}$ gives $Y=\simplify[all,!collectNumbers]{{estal}+{estbe}*{ti}+{adj}}=\var{estal+estbe*ti+adj}=\var{estord}$ orders to the nearest whole number.

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Calculate moving averages, perform linear regression and give seasonal adjustments,

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Norah Storey 4 years, 2 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: `integer`

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Calculate moving averages, perform linear regression and give seasonal adjustments,

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Taxonomy: mathcentre

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Taxonomy: Context

Contributors

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History

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Norah Storey 4 years, 2 months ago

Newcastle University Mathematics and Statistics 9 years, 3 months ago

Error in variable testing condition

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Generated value: integer

Parts

Gap-fill

Generated value: `integer`