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5.06 Further applications

Worksheet
Further applications
1

A hairdressing salon at a local shopping centre records the number of customers per day and the results are shown in the graph below.

Comment on the trend and seasonality of the data.

2

The graph below represents the sales of sausages at a Bunnings store in the four weeks leading up to Christmas:

The table shows the data for the third and fourth weeks:

a

Which day tends to be the busiest for sausage sellers?

b

What type of moving average would best smooth the data?

DaySales
Week 3\text{Fri}162
\text{Sat}484
\text{Sun}600
Week 4\text{Fri}195
\text{Sat}491
\text{Sun}677
3

The following table shows some time series data where t represents time:

a

Calculate the 5 point moving average at

t = 3.

b

Calculate the 4 point centred moving average at t = 3.

t123456
y121422111625
4

The number of children coming to visit Santa Claus on weekdays at a suburban shopping centre is recorded over a 4 week period leading up to Christmas. The following table shows the number of children for each of the weekdays in Week 4:

Day (in Week 4)MonTueWedThuFri
\text{Number of children}75150162205243
a

If the 5 point moving average for Week 4 Tuesday is 151.8, determine x, the actual attendance figure for Week 3 Friday. Round your answer to the nearest whole number.

b

The seasonal indices are shown in the table below. Complete the table by finding the seasonal index for Wednesday.

Day (in Week 4)MonTueWedThuFri
\text{Seasonal indices}40.22\%78.09\%134.05\%162.60\%
c

On Tuesday in Week 4, 150 children came to see Santa Claus. Use the seasonal index to determine the deseasonalised number of visitors for Tuesday Week 4.

d

The deseasonalised number of visitors for Thursday Week 3 is 110.4. Determine the actual number of visitors for this day.

5

An ice-cream store manager recorded the number of ice-creams sold on Friday, Saturday and Sunday over a 4 week period. The given table shows the data for the third and fourth weeks:

a

Calculate the 3 point moving average for Sunday of Week 3. Round your answer to two decimal places.

DaySales
Week 3\text{Fri}145
\text{Sat}492
\text{Sun}604
Week 4\text{Fri}180
\text{Sat}530
\text{Sun}675
b

The seasonal indices for Friday and Sunday are shown in the second table. Calculate the seasonal index for Saturday.

FridaySaturdaySunday
36.11\%143.70\%
c

Complete the third table by calculating the deseasonalised figures of Week 4. Round your answers to the nearest whole number.

DayRaw dataDeseasonalised data
\text{Fri}180
\text{Sat}530
\text{Sun}675
6

A car wash manager recorded the number of cars washed on Friday, Saturday and Sunday over a 4 week period. The tables show the data for the third and fourth weeks, along with the seasonal indices:

DaySales
Week 3\text{Fri}156
\text{Sat}487
\text{Sun}586
Week 4\text{Fri}204
\text{Sat}485
\text{Sun}645
FriSatSun
Seasonal index42.85\%110.35\%146.80\%

Deseasonalise the data and use a linear regression model to predict the actual number of car washes that will be sold on Friday Week 5, to the nearest whole number.

7

A cat boarding kennel records its number of boarders every 4 months (tri-annually) ending in January, May and September. The data of the number of cats, some calculations and the seasonal indices are shown below:

\text{Time}, t\text{Trimester}\text{No. of boarders}\text{Yearly} \\ \text{mean}\text{Percentage} \\ \text{of yearly} \\ \text{mean}\text{Deseasonalised} \\ \text{figure}
2017 1\text{Jan}6450.33 127.2\%48
2\text{May}52 103.3\%54
3\text{Sept}35 C48
2018 4\text{Jan}A50 144.0\%55
5\text{May}45 90.0\%D
6\text{Sept}33 66.0\%46
2019 7\text{Jan}78B 125.1\%59
8\text{May}58 93.1\%61
9\text{Sept}51 81.8\%70

Seasonal indices:

TrimesterJanMaySept
\text{Seasonal index}E0.95450.7245
a

Determine the value A.

b

Determine the value B, to two decimal places.

c

Determine the value C as a percentage rounded to one decimal place.

d

Given the seasonal index for May, determine the value D.

e

Find E, the seasonal index for January.

f

The equation of the least-squares line for the deseasonalised figures against data number is determined to be:y = 2.0333 t + 44.0556

Predict the number of cat boarders for September 2020. Round your answer to the nearest whole number.

g

Comment on the reliability of your prediction.

8

A ballet company performed 4 times a week for four weeks at the Perth Concert Hall which is able to seat 1731 patrons. To break even, the attendance must be more than 60\% of maximum capacity.

The attendance at the daily performances are shown in the table below:

\text{Performance} \\ \text{day}\text{Performance} \\ \text{number }(n)\text{Attendance} \\ \text{number} (A)\text{4CMA}\text{Weekly} \\ \text{mean}\text{Percentage} \\ \text{of weekly} \\ \text{mean}
Week 1\text{Fri}110871223.5 88.8\%
\text{Sat (matinee)}2844 69.0\%
\text{Sat (evening)}317311231.0141.5\%
\text{Sun}41232Y 100.7\%
Week 2\text{Fri}510031192.01153.5 87.0\%
\text{Sat (matinee)}68021172.8 69.5\%
\text{Sat (evening)}717311147.3150.1\%
\text{Sun}810781133.6 93.5\%
Week 3\text{Fri}99531121.11085 87.8\%
\text{Sat (matinee)}107431100.5 68.5\%
\text{Sat (evening)}1116901078.3155.8\%
\text{Sun}129541059.9 87.9\%
Week 4\text{Fri}138991034.5998.5 90.0\%
\text{Sat (matinee)}146501009.6 65.1\%
\text{Sat (evening)}151580158.2\%
\text{Sun}16X 86.6\%
a

What is the break even attendance figure for the ballet company? Round your answer to the nearest whole number.

b

Find the value of:

i

X

ii

Y

c

Given that the seasonal index for the Saturday matinee performance is 68.03\%, calculate the deseasonalised data for the Saturday matinee performance in Week 2. Round your answer to the nearest whole number.

d

Use the average percentage method to calculate the seasonal index for the Friday performances.

e

The attendance data and the moving average data for the four weeks have been graphs on the cartesian plane given.

Describe the trend in attendance data over these four weeks.

f

The ballet company needs 1000 attendees on the average per day to make their performance financially worthwhile.

Should the company continue its performances into a 5th week? Explain your answer.

2
4
6
8
10
12
14
16
n
200
400
600
800
1000
1200
1400
1600
1800
2000
A
g

The equation of the regression line from the 4 point CMA calculated from the raw attendance data for the 4 weeks is:A = - 18.3661 n + 1277.77

Given that the seasonal index for Friday is 88.4\%, predict the attendance figure for Friday of Week 6. Round your answer to the nearest whole number.

h

Comment on the reliability of your prediction.

9

Emma is a university student who runs a small tutoring business. She records the profits three times a year, at the end of April, August and December. The profits are recorded in the table below:

\text{Time} \\ \text{period}\text{Month}\text{Profit }, P(\$1000\text{'s})\text{Yearly} \\ \text{mean}\text{Percentage} \\ \text{of yearly} \\ \text{mean}\text{Deseasonalised} \\ \text{figure}
2017 1\text{Apr}0.92.1 42.86\%D
2\text{Aug}2.4 114.29\%2.2
3\text{Dec}3.0 C2.1
2018 4\text{Apr}1.42.8 50.00\%3.1
5\text{Aug}3.2 114.29\%2.8
6\text{Dec}A 139.29\%2.7
2019 7\text{Apr}2.2B 42.31\%4.9
8\text{Aug}6.0 115.38\%5.2
9\text{Dec}7.3 140.38\%5.2
a

Determine the values of the following to one decimal place:

i

A

ii

B

iii

D

b

Determine the value C as a percentage rounded to two decimal places.

c

Calculate the seasonal index for April. Give your answer as a percentage and round your answer to three decimal places.

d

Consider the two graphs below. Graph 1 shows the profit data (green) graphed along with the deseasonalised data (black). Graph 2 shows the profit data (green) graphed along with a moving average (black).

Graph 1
1
2
3
4
5
6
7
8
9
t
1
2
3
4
5
6
7
8
9
P\left(\$1000\text{'s}\right)
Graph 2
1
2
3
4
5
6
7
8
9
t
1
2
3
4
5
6
7
8
9
P\left(\$1000\text{'s}\right)

Which method of smoothing seems to be the most appropriate for this situation?

e

Which type of moving average should Emma use to smooth the data?

f

The table below shows the 3 point moving averages for the profit data:

\text{Time}\\ \text{period}\text{Month}\text{Profit }(\$1000\text{'s})\text{3MA}
2017 1\text{Apr}0.9
2\text{Aug}2.42.10
3\text{Dec}3.02.27
2018 4\text{Apr}1.4B
5\text{Aug}3.22.80
6\text{Dec}3.82.97
2019 7\text{Apr}A3.90
8\text{Aug}6.05.07
9\text{Dec}7.3

Determine the value of:

i
A
ii
B
g

Emma decides to use a 3 point moving average to smooth the data in order to forecast. The equation of the least-squares regression line for the moving average data is: A = 0.8657 t + 0.7679

What will her prediction be for the tutoring profits in August 2020?

h

Comment on the reliability of Emma's prediction.

10

An office manager recorded the number of coffee pods used daily by staff over a 4 week period. The data and seasonal indices are recorded in the tables below:

\text{Weekday}\text{Day, }d\text{Number of}\\ \text{coffee pods }\text{Weekly } \\ \text{mean}\text{Percentage of}\\ \text{weekly mean }
Week 1\text{Mon}1134154.2 86.9\%
\text{Tues}2154 99.87\%
\text{Wed}314392.74\%
\text{Thurs}4165 107.00\%
\text{Fri}5175 113.49\%
Week 2\text{Mon}6137157.6 86.93\%
\text{Tues}7158 100.25\%
\text{Wed}814592.01\%
\text{Thurs}9169 107.23\%
\text{Fri}10179 113.58\%
Week 3\text{Mon}11140159.687.72\%
\text{Tues}12160 100.25\%
\text{Wed}1314792.11\%
\text{Thurs}14171 107.14\%
\text{Fri}15180 112.78\%
Week 4\text{Mon}16145164.2 88.31\%
\text{Tues}17163 99.27\%
\text{Wed}1815191.96\%
\text{Thurs}19176 107.19\%
\text{Fri}20186 113.28\%
DayMondayTuesdayWednesdayThursdayFriday
\text{Seasonal index}0.87460.99910.92211.07141.1328

Use a least squares regression line to predict the actual number of coffee pods used for Tuesday of Week 6.

11

A Bunnings events manager recorded the number of sausages sold by not-for-profit fundraisers on Friday, Saturday and Sunday over a 4 week period. The data is recorded in the table below, together with some calculations:

\text{Weekday}\text{Day, }d\text{Number of }\\ \text{sausages}\text{Weekly} \\ \text{mean}\text{Percentage } \\ \text{weekly mean}
Week 1\text{Fri}1136375.0036.27\%
\text{Sat}2443118.13\%
\text{Sun}3546145.60\%
Week 2\text{Fri}4140377.3337.10\%
\text{Sat}5A116.08\%
\text{Sun}6554146.82\%
Week 3\text{Fri}7132378.67C
\text{Sat}8443119.89\%
\text{Sun}9550145.25\%
Week 4\text{Fri}10135B35.75\%
\text{Sat}11441116.77\%
\text{Sun}12557147.48\%
a

Determine the value of:

i

A

ii

B

iii

C

b

Complete the given table by calculating the seasonal index for Sunday.

c

132 sausages were sold on Friday of Week 3. Use the seasonal index to find the deseasonalised number of sausages for Friday of Week 3 to the nearest whole number.

DaySeasonal index
\text{Friday}0.3599
\text{Saturday}1.1772
\text{Sunday}
d

The data is deseasonalised and the equation of the least-squares line for the deseasonalised number of sausages is:\text{Deseasonalised number of sausages} = 0.0105 d + 377.1818

Describe the trend in the number of sausages used over time.

e

Predict the actual number of sausages sold on Saturday of Week 5.

f

Comment on the reliability of your prediction.

12

A hairdressing salon at a local shopping centre records the number of customers per day. The data is shown in the table below and the average percentage method is used to deseasonalise the data:

\text{Weekday}\text{Day}, d\text{Number of} \\ \text{customers}\text{Weekly} \\ \text{mean}\text{Percentage} \\ \text{weekly mean}\text{Deseasonalised} \\ \text{data}
Week 1\text{Mon}12139.47\%41.6
\text{Tues}24482.71\%41.6
\text{Wed}34853.290.23\%51.2
\text{Thurs}4A122.18\%57.8
\text{Fri}588165.41\%56.5
Week 2\text{Mon}63352.72\%65.4
\text{Tues}75791.05\%65.1
\text{Wed}861B97.44\%65.1
\text{Thurs}970111.82\%62.3
\text{Fri}1092146.96\%59.1
Week 3\text{Mon}113853.37\%75.3
\text{Tues}126388.48\%71.9
\text{Wed}136671.2C70.4
\text{Thurs}1477108.15\%68.5
\text{Fri}15112157.30\%71.9
Week 4\text{Mon}164456.27\%87.2
\text{Tues}176988.24\%D
\text{Wed}187478.295.63\%78.9
\text{Thurs}1984107.42\%74.7
\text{Fri}20120153.45\%77.0
a

Determine the values of the following. Round your answers to one decimal place if necessary.

i

A

ii

B

iii

D

b

Determine the value C as a percentage rounded to two decimal places.

c

State the equation of the least-squares regression line for the deseasonalised data in the form: N = a d + b with a and b rounded to four decimal places.

d

Calculate the seasonal index for Friday. Give your answer as a percentage.

e

The regression model for the deseasonalised number of customers is given by: N = 1.7051 d + 48.5363 Predict the number of customers for Friday of Week 5. Round your answer to the nearest whole number.

f

Comment on the reliability of your prediction.

13

The number of children coming to visit Santa Claus on weekdays at a suburban shopping centre is recorded over a 4 week period leading up to Christmas. The data is displayed on the graph below:

The data for the next 5 days is shown in the following table:

a

Complete the time series plot by including this additional information.

b

What type of moving average would best smooth this data?

c

Explain why this moving average is the best fit for this data.

No. of Children
Wk 4 - Mon75
Wk 4 - Tue150
Wk 4 - Wed162
Wk 4 - Thu174
Wk 4 - Fri180
14

The number of visitors to the zoo on weekdays during the summer school holidays is recorded over a 4 week period. The data is recorded on the graph below:

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
d
20
40
60
80
100
120
140
160
180
200
220
240
N

The equation of the least-squares line for the data is: N= 6.007d + 41.3259Where N is the number of visitors and d is the day number with d=1 for Week 1 Monday, d=2 for Week 1 Tuesday, d=6 for Week 2 Monday, etc.

a

Graph this line and the above data on the same number plane.

b

Describe on the overall trend of the data.

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Outcomes

3.2.1.2

describe time series plots by identifying features such as trend (long-term direction), seasonality (systematic, calendar-related movements) and irregular fluctuations (unsystematic, short-term fluctuations), and recognise when there are outliers, e.g. one-off unanticipated events

3.2.2.1

smooth time series data by using a simple moving average, including the use of spreadsheets to implement this process

3.2.2.2

calculate seasonal indices by using the average percentage method

3.2.2.3

deseasonalise a time series by using a seasonal index, including the use of spreadsheets to implement this process

3.2.2.4

fit a least-squares line to model long-term trends in time series data, using appropriate technology

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