The time series plot below shows the gold medal-winning height for the women's high jump, \(\textit{Wgold}\), in metres, for each Olympic year, year, from 1952 to 1988. A five-median smoothing process will be used to smooth the time series plot above. The first two points have been placed on the graph with crosses (X) and joined by a dashed line (---). --- 0 WORK AREA LINES (style=lined) --- --- 2 WORK AREA LINES (style=lined) --- a. b. \(\text{Random fluctuations, increasing trend.}\) a. \(\text{Medians are:}\) \(\textbf{1960}: 1.67, 1.76, \colorbox{lightblue}{1.82}, 1.85, 1.90\) \(\textbf{1964}: 1.76, 1.82, \colorbox{lightblue}{1.85}, 1.90, 1.92\) \(\textbf{1968}: 1.82, 1.85, \colorbox{lightblue}{1.90}, 1.92, 1.93\) \(\textbf{1972}: 1.82, 1.90, \colorbox{lightblue}{1.92}, 1.93, 1.97\) \(\textbf{1976}: 1.82, 1.92, \colorbox{lightblue}{1.93}, 1.97, 2.02\) \(\textbf{1980}: 1.92, 1.93, \colorbox{lightblue}{1.97}, 2.01, 2.02\) b. \(\text{Time series shows random fluctuations with an increasing trend.}\)
Data Analysis, GEN1 2024 VCAA 13-14 MC
A school runs an orientation program for new staff each January.
The time series plot below shows the number of new staff, \(new\), for each year, \(year\), from 2011 to 2022 (inclusive).
Part A
The time series is smoothed using seven-median smoothing.
The smoothed value of \(new\) for the \(year\) 2016 is
- 10
- 11
- 12
- 13
Part B
The number of new staff in 2023 is added to the total number of new staff from the previous 12 years. For these 13 years, the mean number of new staff is 11 .
The number of new staff in 2023 is
- 11
- 16
- 17
- 19
Data Analysis, GEN1 2023 VCAA 13-14 MC
The following graph shows a selection of winning times, in seconds, for the women's 800 m track event from various athletic events worldwide. The graph shows one winning time for each calendar year from 2000 to 2022.
Question 13
The time series is smoothed using seven-median smoothing.
The smoothed value for the winning time in 2006, in seconds, is closest to
- 116.0
- 116.4
- 116.8
- 117.2
- 117.6
Question 14
The median winning time, in seconds, for all the calendar years from 2000 to 2022 is closest to
- 116.8
- 117.2
- 117.6
- 118.0
- 118.3
CORE, FUR1 2021 VCAA 13 MC
The time series plot below shows the points scored by a basketball team over 40 games.
The nine-median smoothed points scored for game number 10 is closest to
- 102
- 108
- 110
- 112
- 117
CORE, FUR1 2020 VCAA 19-20 MC
The time series plot below displays the number of airline passengers, in thousands, each month during the period January to December 1960.
Part 1
During 1960, the median number of monthly airline passengers was closest to
- 461 000
- 465 000
- 471 000
- 573 000
- 621 000
Part 2
During the period January to May 1960, the total number of airline passengers was 2 160 000.
The five-mean smoothed number of passengers for March 1960 is
- 419 000
- 424 000
- 430 000
- 432 000
- 434 000
CORE, FUR1-NHT 2019 VCAA 14 MC
The time series plot below shows the daily number of visitors to a historical site over a two-week period.
This time series plot is to be smoothed using seven-median smoothing.
The smoothed number of visitors on day 4 is closest to
- 120
- 140
- 145
- 150
- 160
CORE, FUR1 2019 VCAA 13-14 MC
The time, in minutes, that Liv ran each day was recorded for nine days.
These times are shown in the table below.
The time series plot below was generated from this data.
Part 1
Both three-median smoothing and five-median smoothing are being considered for this data.
Both of these methods result in the same smoothed value on day number
- 3
- 4
- 5
- 6
- 7
Part 2
A least squares line is to be fitted to the time series plot shown above.
The equation of this least squares line, with day number as the explanatory variable, is closest to
- day number = 23.8 + 2.29 × time
- day number = 28.5 + 1.77 × time
- time = 23.8 + 1.77 × day number
- time = 23.8 + 2.29 × day number
- time = 28.5 + 1.77 × day number
Show Worked Solution
`text(Part 1)`
`text{Add 3-median (dots) and 5-median (Δ) smoothing to the plot:}`
`=> E`
`text(Part 2)`
`text(time) = 28.5 + 1.77 xx text(day number)\ \ \ text{(by CAS)}`
`=> E`
CORE, FUR1 2017 VCAA 13-15 MC
The wind speed at a city location is measured throughout the day.
The time series plot below shows the daily maximum wind speed, in kilometres per hour, over a three-week period.
Part 1
The time series is best described as having
- seasonality only.
- irregular fluctuations only.
- seasonality with irregular fluctuations.
- a decreasing trend with irregular fluctuations.
- an increasing trend with irregular fluctuations.
Part 2
The seven-median smoothed maximum wind speed, in kilometres per hour, for day 4 is closest to
- `22`
- `26`
- `27`
- `30`
- `32`
Part 3
The table below shows the daily maximum wind speed, in kilometres per hour, for the days in week 2.
A four-point moving mean with centring is used to smooth the time series data above.
The smoothed maximum wind speed, in kilometres per hour, for day 11 is closest to
- `22`
- `24`
- `26`
- `28`
- `30`
Show Worked Solution
`text(Part 1)`
`text(The time series plot shows no obvious trend and)`
`text(is over too short a period to show seasonality.)`
`=> B`
`text(Part 2)`
`text(Consider the 7 values where day 4 is the middle)`
`text(data point.)`
`text(By inspection of the graph, the 4th highest point = 30.)`
`=> D`
`text(Part 3)`
`text(Mean for Day 9 – 12)`
`= (22 + 19 + 22 + 43)/4 = 26.5`
`text(Mean for Day 10 – 13)`
`= (19 + 22 + 43 + 37)/4 = 30.25`
`:. 4text(-point moving mean with centring)`
`= (26.5 + 30.25)/2`
`= 28.375`
`=> D`
CORE, FUR2 2016 VCAA 4
The time series plot below shows the minimum rainfall recorded at the weather station each month plotted against the month number (1 = January, 2 = February, and so on).
Rainfall is recorded in millimetres.
The data was collected over a period of one year.
- Five-median smoothing has been used to smooth the time series plot above.
The first four smoothed points are shown as crosses (×).
Complete the five-median smoothing by marking smoothed values with crosses (×) on the time series plot above. (2 marks)
The maximum daily rainfall each month was also recorded at the weather station.
The table below shows the maximum daily rainfall each month for a period of one year.
The data in the table has been used to plot maximum daily rainfall against month number in the time series plot below.
- Two-mean smoothing with centring has been used to smooth the time series plot above.
The smoothed values are marked with crosses (×).
Using the data given in the table, show that the two-mean smoothed rainfall centred on October is 157.25 mm. (2 marks)
Show Worked Solution
| a. |  |
♦ Mean mark of both Parts (a) and (b) was 49%.
MARKER’S COMMENT: Use the accurate table data when available. Reading values from the graph will cause inaccuracies.
MARKER’S COMMENT: Use the accurate table data when available. Reading values from the graph will cause inaccuracies.
| b. | `text(Mean)\ _text(Sep-Oct)` | `= (124 + 140)/2` |
| `= 132\ text(mm)` | ||
| `text(Mean)\ _text(Oct-Nov)` | `= (140 + 225)/2` | |
| `= 182.5\ text(mm)` |
`:.\ text{Two mean (smoothed) for October}`
`= (132 +182.5)/2`
`= 157.25\ text(mm … as required)`
CORE, FUR2 2011 VCAA 3
The following time series plot shows the average age of women at first marriage in a particular country during the period 1915 to 1970.
- Use this plot to describe, in general terms,the way in which the average age of women at first marriage in this country has changed during the period 1915 to 1970. (1 mark)
During the period 1986 to 2006, the average age of men at first marriage in a particular country indicated an increasing linear trend, as shown in the time plot below.
A three-median line could be used to model this trend.
- On the graph above
- clearly mark with a cross (×) the three points that would be used to fit a three-median line to this time series plot (2 marks)
- draw in the three-median line. (1 mark)
Show Answers Only
- `text(Constant between 1915 and 1935 and then)`
`text(decreasing between 1935 and 1970.)` -
Show Worked Solution
a. `text(The average age of women at first marriage was fairly)`
`text(constant between 1915 and 1935, and then decreased)`
`text(between 1935 and 1970.)`
b.i. & ii.
CORE, FUR1 2010 VCAA 12 MC
The time series plot below shows the number of calls each month to a call centre over a twelve-month period.
The plot is to be smoothed using five-point median smoothing.
The smoothed number of calls for month number 10 is closest to
A. `358`
B. `364`
C. `371`
D. `375`
E. `377`
CORE, FUR1 2015 VCAA 12 MC
The time series plot below charts the number of calls per year to a computer help centre over a 10-year period.
Using five-median smoothing, the smoothed number of calls in year 6 was closest to
A. `3500`
B. `3700`
C. `3800`
D. `4000`
E. `4200`
CORE, FUR1 2014 VCAA 13 MC
The time series plot below shows the hours of sunshine per day at a particular location for 16 consecutive days.
The three median method is used to fit a trend line to the data.
The slope of this trend line will be closest to
A. `–0.7`
B. `–0.2`
C. `0.0`
D. `0.2`
E. `0.7`
CORE, FUR1 2009 VCAA 13 MC
The time series plot below shows the growth in Internet use (%) in a country from 1989 to 1997 inclusive.
If a three-median line is fitted to the data it would show that, on average, the increase in Internet use per year was closest to
A. `0.33text(%)`
B. `0.36text(%)`
C. `0.41text(%)`
D. `0.45text(%)`
E. `0.49text(%)`
CORE, FUR1 2012 VCAA 9 MC
The time series plot below shows the number of days that it rained in a town each month during 2011.
Using five-median smoothing, the smoothed time series plot will look most like
