Further applications can require us to apply our knowledge across the different topics in this chapter and recognise the appropriate techniques to apply. Applications may entail use of the following skills:

Recognising a cyclical pattern in a time series graph and stating the appropriate moving average.
Recognising an outlier or fluctuation in a time series graph.
Plotting data points for a time series graph.
Smoothing data using an odd or even moving average .
Using a moving average formula to find an unknown data value.
Using the average percentage method to calculate seasonal indices .
Smoothing data by using seasonal indices to deseasonalise data.
Using formula for cycle mean, percentage of cycle mean and seasonal index to find a missing value in a table of data .
Using a calculator to determine the equation of a least-squares regression line for time series data that has been smoothed either using the moving average data OR deseasonalised data.
Predicting a future value by using the least-squares regression line formula and then multiplying the result by the seasonal index.
Commenting on the reliability of the prediction by observing how close the time period is to the existing time series data (within one cycle is considered reliable).

Examples

Example 1

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

t	1	2	3	4	5	6
y	12	14	22	11	16	25

Calculate the 5 point moving average at t=3.

Worked Solution

Create a strategy

Find the mean of the data values for the 2 previous, current and 2 next time periods.

Apply the idea

We need to find the average of the y-values for t=1 to t=5.

\displaystyle \text{5MA}	\displaystyle =	\displaystyle \dfrac{12+14+22+11+16}{5}	Find the mean of the data values
	\displaystyle =	\displaystyle 15	Evaluate

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

Worked Solution

Create a strategy

Use formula: \text{4CMA}=\dfrac{0.5a+b+c+d+0.5e}{4}.

Apply the idea

We should use the y-values for t=1 to t=5 in the formula.

\displaystyle \text{4CMA}	\displaystyle =	\displaystyle \dfrac{0.5\times 12 + 14 + 22 + 11 + 0.5 \times 16}{4}	Substitute the 5 data values
	\displaystyle =	\displaystyle 15.25	Evaluate

Example 2

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{Data number}\\ (d)	\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}	64	50.33	127.2\%	48
	2	\text{May}	52		103.3\%	54
	3	\text{Sept}	35		C	48

2018	4	\text{Jan}	A	50	144.0\%	55
	5	\text{May}	45		90.0\%	D
	6	\text{Sept}	33		66.0\%	46

2019	7	\text{Jan}	78	B	125.1\%	59
	8	\text{May}	58		93.1\%	61
	9	\text{Sept}	51		81.8\%	70

Determine the value A.

Worked Solution

Create a strategy

Form an equation using the yearly mean and solve for A.

Apply the idea

Use the yearly mean for 2018 to create an equation containing A:

\displaystyle \dfrac{A +45 +33}{3}	\displaystyle =	\displaystyle 50	Equate the mean to the average of the data
\displaystyle \dfrac{A+78}{3}	\displaystyle =	\displaystyle 50	Simplify the numerator
\displaystyle A+78	\displaystyle =	\displaystyle 150	Multiply both sides by 3
\displaystyle A	\displaystyle =	\displaystyle 72	Subtract 78 from both sides

Determine the value B. Round your answer to two decimal places.

Worked Solution

Create a strategy

Find the average of the number of boarders each trimester in 2019.

Apply the idea

\displaystyle B	\displaystyle =	\displaystyle \dfrac{78+58+51}{3}	Find the average
	\displaystyle =	\displaystyle 62.33	Evaluate

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

Worked Solution

Create a strategy

Divide the raw data score by the yearly mean and multiply by 100\%.

Apply the idea

C is the percentage for September 2017. The number of boarders for September 2017 was 35 and the yearly mean for 2017 was 50.33.

\displaystyle C	\displaystyle =	\displaystyle \dfrac{35}{50.33}\times 100\%	Divide the data by the mean and multiply by 100\%
	\displaystyle \approx	\displaystyle 69.5\%	Evaluate and round

If the seasonal index for May is 0.9545, determine the value D. Round your answer to the nearest whole number.

Worked Solution

Create a strategy

Use formula: \text{Deseasonalised data} = \dfrac{\text{raw data}}{\text{seasonal index}}.

Apply the idea

The number of boarders for May 2018 was 45.

\displaystyle D	\displaystyle =	\displaystyle \dfrac{45}{0.9545}	Divide the raw data by the seasonal index
	\displaystyle \approx	\displaystyle 47	Evaluate and round

Example 3

	\text{Data number}\\(d)	\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}	64	50.33	127.2\%	48
	2	\text{May}	52		103.3\%	54
	3	\text{Sept}	35		69.5\%	48

2018	4	\text{Jan}	72	50	144.0\%	55
	5	\text{May}	45		90.0\%	47
	6	\text{Sept}	33		66.0\%	46

2019	7	\text{Jan}	78	62.33	125.1\%	59
	8	\text{May}	58		93.1\%	61
	9	\text{Sept}	51		81.8\%	70

Seasonal indices:

Trimester	Jan	May	Sept
\text{Seasonal index}	1.321	0.9545	0.7245

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

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

Worked Solution

Create a strategy

Use the given equation and then multiply the result by the September seasonal index.

Apply the idea

For September 2020, d=9+3=12.

\displaystyle y	\displaystyle =	\displaystyle 2.0333 \times 12 + 44.0556	Substitute d=12
	\displaystyle =	\displaystyle 68.4516	Evaluate

To find the seasonalised prediction for September 2020, we should multiply this result by the seasonal index for September which is 0.724.

\displaystyle \text{Predicted value}	\displaystyle =	\displaystyle 68.4516 \times 0.724	Multiply by the seasonal index
	\displaystyle =	\displaystyle 50	Evaluate the product

Comment on the reliability of your prediction.

It's reliable because the prediction being was made within one cycle of the data.

It's unreliable because the prediction being was made beyond one cycle of the data.

Worked Solution

Create a strategy

Look if the prediction made in part (a) is within or beyond one cycle of the data.

Apply the idea

The answer in part (a) is within one year of the given data, so it is within one cycle. The correct option is A.

Idea summary

Further applications questions may require the following knowledge and skills:

Time series graphs.
Smoothing data and moving averages.
Seasonal indices and deseasonalising data.
Cycle mean and percentage of cycle mean.
Least-squares regression line for time series data that has been smoothed.
Predicting a future value using the least-squares regression line.
Commenting on the reliability of the prediction.

Outcomes

ACMGM088

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; for example, one-off unanticipated events

ACMGM089

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

ACMGM090

calculate seasonal indices by using the average percentage method

ACMGM091

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

ACMGM092

fit a least-squares line to model long-term trends in time series data

5.06 Further applications

Ideas

Further applications

Examples

Example 1

Example 2

Example 3

Outcomes

ACMGM088

ACMGM089

ACMGM090

ACMGM091

ACMGM092

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