Examination questions on this topic could include:
- 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 exisiting time series data (within one cycle is considered reliable).
Practice examination style questions are provided in the accompanying exercise.
The Year 12 Applications 2019 formula sheet contains the following formula that may be useful.
| Least-squares line |
$\hat{y}=a+bx$^y=a+bx where $y$y is the response variable, and $x$x is the explanatory variable. $\hat{y}=a+bt$^y=a+bt where $y$y is the response variable, and $t$t is time (the explanatory variable). |
| $\text{Deseasonalised value}=\frac{\text{Actual value}}{\text{Seasonal index}}$Deseasonalised value=Actual valueSeasonal index | |
Note that predicted values are denoted by $y$y-hat ($\hat{y}$^y), however, exams commonly don't use this notation, and often drop the hat and just use $y$y.
Practice questions
question 1
The table below shows some time series data where $t$t represents time.
| $t$t | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 |
|---|---|---|---|---|---|---|
| $y$y | $12$12 | $14$14 | $22$22 | $11$11 | $16$16 | $25$25 |
Calculate the $5$5 point moving average at $t=3$t=3.
Calculate the $4$4 point centred moving average at $t=3$t=3.
question 2
A cat boarding kennel records its number of boarders every $4$4 months (tri-annually) ending at January, May and September. The data of the cat numbers, together with some calculations are shown in the table below.
| Year | Data number ($d$d) | Trimester | Number of boarders | Yearly mean | Percent of yearly mean | Deseasonalised figure |
|---|---|---|---|---|---|---|
| 2017 | $1$1 | Jan | $64$64 | $50.33$50.33 | $127.2%$127.2% | $48$48 |
| $2$2 | May | $52$52 | $103.3%$103.3% | $54$54 | ||
| $3$3 | Sept | $35$35 | $C$C | $48$48 | ||
| 2018 | $4$4 | Jan | $A$A | $50$50 | $144.0%$144.0% | $55$55 |
| $5$5 | May | $45$45 | $90.0%$90.0% | $D$D | ||
| $6$6 | Sept | $33$33 | $66.0%$66.0% | $46$46 | ||
| 2019 | $7$7 | Jan | $78$78 | $B$B | $125.1%$125.1% | $59$59 |
| $8$8 | May | $58$58 | $93.1%$93.1% | $61$61 | ||
| $9$9 | Sept | $51$51 | $81.8%$81.8% | $70$70 |
Determine the value $A$A.
Determine the value $B$B.
Round your answer to two decimal places.
Determine the value $C$C.
Give your answer as a percentage rounded to one decimal place.
If the seasonal index for May is $0.9545$0.9545, determine the value $D$D.
Round your answer to the nearest whole number.
question 3
A cat boarding kennel records its number of boarders every $4$4 months (tri-annually) ending at January, May and September.
The seasonal indices are shown in the table below. Complete the table by finding the seasonal index for January.
Trimester Jan May Sept Seasonal indices $\editable{}$ $0.9545$0.9545 $0.7245$0.7245
question 4
A cat boarding kennel records its number of boarders every $4$4 months (tri-annually) ending at January, May and September. The data of the cat numbers, together with some calculations are shown in the table below.
| Year | Data number ($d$d) | Trimester | Number of boarders | Yearly mean | Percent of yearly mean | Deseasonalised figure |
|---|---|---|---|---|---|---|
| 2017 | $1$1 | Jan | $64$64 | $50.33$50.33 | $127.2%$127.2% | $48$48 |
| $2$2 | May | $52$52 | $103.3%$103.3% | $54$54 | ||
| $3$3 | Sept | $35$35 | $69.5%$69.5% | $48$48 | ||
| 2018 | $4$4 | Jan | $72$72 | $50$50 | $144.0%$144.0% | $55$55 |
| $5$5 | May | $45$45 | $90.0%$90.0% | $47$47 | ||
| $6$6 | Sept | $33$33 | $66.0%$66.0% | $46$46 | ||
| 2019 | $7$7 | Jan | $78$78 | $62.33$62.33 | $125.1%$125.1% | $59$59 |
| $8$8 | May | $58$58 | $93.1%$93.1% | $61$61 | ||
| $9$9 | Sept | $51$51 | $81.8%$81.8% | $70$70 |
The equation of the least-squares line for the deseasonalised figures against data number is determined to be $y=2.0333d+44.0556$y=2.0333d+44.0556
Predict the number of cat boarders for September 2020. Round your answer to the nearest whole number.
Comment on the reliability of your prediction.
It's reliable because the prediction was made within one cycle of the data.
AIt's unreliable because the prediction was made beyond one cycle of the data.
B