From the beginning of 2016, the number of new houses built in the suburb of Woodford was recorded and figures are released every four months.
The table contains the data from the beginning of 2016 to the end of 2019:
Calculate the mean number of houses built, to two decimal places, for each of the following years:
2016
2017
2018
Hence, calculate the values of the following to two decimal places:
x
y
z
| Time period | Houses built | Percentage of yearly mean |
|---|---|---|
| \text{April 2016} | 6 | 60\% |
| \text{August 2016} | 21 | x |
| \text{December 2016} | 3 | 30\% |
| \text{April 2017} | 11 | y |
| \text{August 2017} | 24 | 180\% |
| \text{December 2017} | 5 | 37.5\% |
| \text{April 2018} | 15 | 84.91\% |
| \text{August 2018} | 25 | 141.51\% |
| \text{December 2018} | 13 | z |
| \text{April 2019} | 15 | 77.59\% |
| \text{August 2019} | 28 | 144.83\% |
| \text{December 2019} | 15 | 77.59\% |
The local police station records the number of speeding fines issued each quarter. The table shows the data for each quarter from 2016 to 2018:
Calculate the mean number of speeding tickets issued, to two decimal places, for each of the following years:
2016
2017
2018
Hence, calculate the values of the following to two decimal places:
x
y
z
| Time period | Speeding fines | Percentage of yearly mean |
|---|---|---|
| \text{March 2016} | 105 | 106.06\% |
| \text{June 2016} | 91 | x |
| \text{September 2016} | 101 | 102.02\% |
| \text{December 2016} | 99 | 100\% |
| \text{March 2017} | 101 | y |
| \text{June 2017} | 83 | 89.01\% |
| \text{September 2017} | 96 | 102.95\% |
| \text{December 2017} | 93 | 99.73\% |
| \text{March 2018} | 99 | 108.2\% |
| \text{June 2018} | 82 | 89.62\% |
| \text{September 2018} | 94 | 102.73\% |
| \text{December 2018} | 91 | z |
What is the purpose of deseasonalising data?
Explain the meaning of the following seasonal indices in context:
The seasonal index for heater sales in winter is 1.30.
The seasonal index for Wednesday's earnings is 0.81.
The seasonal index for the sales of cold drinks in a shop in January is 1.6.
To correct the January sales of cold drinks for seasonality, what calculation should be applied?
A company sells central heating systems. The following table shows the quarterly seasonal indices for sales in the last three quarters of a year. One year represents one cycle.
| Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |
|---|---|---|---|---|
| Seasonal Index | 1.30 | 1.45 | 0.58 |
Calculate the seasonal index for Quarter 1.
The deseasonalised sales for Quarter 1 in 2018 were \$2\,500\,000 million. Calculate the actual sales in dollars.
A pizza shop that is open seven days a week has the following seasonal indices:
| Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday | |
|---|---|---|---|---|---|---|---|
| Seasonal Index | 0.4 | 0.3 | 0.5 | 0.8 | 1.9 | 1.2 |
The index for Friday has not been recorded. Calculate the missing index.
Calculate the seasonal index for Winter for the following data:
| Summer | Autumn | Winter | Spring | |
|---|---|---|---|---|
| Seasonal Index | 1.20 | 0.92 | 1.48 |
The following table contains the seasonal indices for the monthly sales of spring water in a particular supermarket. One year represents one cycle:
Calculate the seasonal index for February.
If the actual sales figure for June was \$108\,000, calculate the deseasonalised figure.
If the deseasonalised figure for December is \$110\,000, calculate the actual sales for December.
| Month | Seasonal index |
|---|---|
| \text{January} | 1.00 |
| \text{February} | |
| \text{March} | 1.00 |
| \text{April} | 1.00 |
| \text{May} | 0.95 |
| \text{June} | 0.80 |
| \text{July} | 0.75 |
| \text{August} | 0.95 |
| \text{September} | 0.95 |
| \text{October} | 1.10 |
| \text{November} | 1.10 |
| \text{December} | 1.15 |
The following table displays the quarterly newspaper sales (in thousands of dollars) of a corner store. Also shown are the seasonal indices for the four quarters:
| Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |
|---|---|---|---|---|
| Sales | 1058 | 1852 | 1650 | |
| Seasonal Index | 0.9 | 0.8 | 1.4 |
Calculate the seasonal index for Quarter 4.
Which quarter is peak season for newspaper sales? Explain your answer.
Calculate the deseasonalised sales figure for Quarter 2.
Calculate the deseasonalised sales figure for Quarter 3.
The deseasonalised sales figure for Quarter 1 is 1238. Calculate the actual sales figure for Quarter 1.
The number of waiters employed by a restaurant chain in each quarter of one year, along with the quarterly seasonal indices are displayed in the following table:
| Quarter 1 | Quarter 2 | Quarter 3 | Quarter 4 | |
|---|---|---|---|---|
| Number of waiters | 197 | 144 | 83 | 170 |
| Seasonal Index | 1.31 | 0.57 | 1.08 |
Calculate the seasonal index for the second quarter.
The seasonal index for Quarter 1 is 1.31. Explain the meaning of this figure.
Deseasonalise the data for each quarter. Round your answers to the nearest whole number.
The seasonal indices for the first 11 months of the year for sales in a sporting equipment store are shown in the table. The full year represents one cycle.
Calculate the seasonal index for December.
In May, the store sold \$210\,000 worth of sporting equipment. Calculate the deseasonalised value of the May sales.
In February, the deseasonalised value of sales was \$200\,000. Calculate the actual sales figures for February.
| Month | Seasonal Index |
|---|---|
| \text{Jan} | 1.23 |
| \text{Feb} | 0.99 |
| \text{Mar} | 1.12 |
| \text{Apr} | 1.11 |
| \text{May} | 0.89 |
| \text{Jun} | 0.96 |
| \text{Jul} | 0.86 |
| \text{Aug} | 0.73 |
| \text{Sep} | 0.77 |
| \text{Oct} | 0.96 |
| \text{Nov} | 1.10 |
| \text{Dec} |
The table shows the number of new gaming apps released each quarter, from the beginning of 2016 through to the end of 2018.
Answer the following, rounding your answers to two decimal places.
Calculate the seasonal index for the first quarter (March).
Deseasonalise the data for March 2018.
Calculate the seasonal index for the second quarter (June).
Calculate the seasonal index for the third quarter (September).
Calculate the seasonal index for the fourth quarter (December).
Seasonally adjust the data for December 2017.
Calculate the mean of the seasonal indices.
| Month | Data | Seasonal Index |
|---|---|---|
| \text{March 2016} | 43 | 0.77 |
| \text{June 2016} | 45 | 0.81 |
| \text{September 2016} | 54 | 0.97 |
| \text{December 2016} | 81 | 1.45 |
| \text{March 2017} | 51 | 0.82 |
| \text{June 2017} | 50 | 0.8 |
| \text{September 2017} | 60 | 0.96 |
| \text{December 2017} | 89 | 1.42 |
| \text{March 2018} | 57 | 0.81 |
| \text{June 2018} | 52 | 0.74 |
| \text{September 2018} | 69 | 0.98 |
| \text{December 2018} | 103 | 1.47 |
The adjacent table displays data on the number of holidays taken by a class of Year 10 students in Belgium. Data is recorded three times per year.
Answer the following, rounding your answers to two decimal places.
Which season is 'peak season'?
Calculate the seasonal index for the April time period.
Hence deseasonalise the data for April 2018.
Calculate the seasonal index for the August time period.
Seasonally adjust the data for August 2016.
Calculate the seasonal index for the December season.
Seasonally adjust the data for December 2017.
Calculate the mean of the seasonal indices.
| Time period | Data | Percentage of yearly mean |
|---|---|---|
| \text{April 2016} | 6 | 60\% |
| \text{August 2016} | 21 | 210\% |
| \text{December 2016} | 3 | 30\% |
| \text{April 2017} | 11 | 83\% |
| \text{August 2017} | 24 | 180\% |
| \text{December 2017} | 5 | 38\% |
| \text{April 2018} | 15 | 25\% |
| \text{August 2018} | 25 | 142\% |
| \text{December 2018} | 13 | 74\% |
| \text{April 2019} | 15 | 78\% |
| \text{August 2019} | 28 | 145\% |
| \text{December 2019} | 15 | 78\% |
Every four months Neil records the growth of his bean plant, starting with a new plant every year. The data provided is from the beginning of 2016 to the end of 2019.
Answer the following, rounding your answers to two decimal places.
Calculate the value of a in the table.
Calculate the value of b in the table.
If the mean for 2017 is 92.2, calculate the value of c.
Calculate the seasonal component for April.
Seasonally adjust the data for April 2019.
| \text{Time period} | \text{Growth} \\ \text{(in cm)} | \text{Proportion} \\ \text{of yearly mean} |
|---|---|---|
| \text{April 2016} | 95.6 | 0.99 |
| \text{August 2016} | 106.7 | a |
| \text{December 2016} | 87.8 | 0.91 |
| \text{April 2017} | c | 0.99 |
| \text{August 2017} | 101.2 | 1.1 |
| \text{December 2017} | 84.1 | 0.91 |
| \text{April 2018} | 86.3 | 1.01 |
| \text{August 2018} | 93.6 | 1.09 |
| \text{December 2018} | 77.3 | 0.9 |
| \text{April 2019} | 76.1 | 0.99 |
| \text{August 2019} | 83.4 | b |
| \text{December 2019} | 71.8 | 0.93 |
Consider the data provided for each of the following time periods from 2016 to 2018:
Answer the following, rounding your answers to two decimal places.
Calculate the value of a in the table.
Calculate the value of b in the table.
If the mean for 2018 is 86.075, calculate the value of c.
Calculate the seasonal component for the third quarter.
Deseasonalise the data for September 2017.
| \text{Time period} | \text{Data} | \text{Proportion} \\ \text{of yearly mean} |
|---|---|---|
| \text{March 2016} | 108.2 | 1.12 |
| \text{June 2016} | 80.9 | 0.84 |
| \text{September 2016} | 102.4 | a |
| \text{December 2016} | 94.6 | 0.98 |
| \text{March 2017} | 95.3 | 1.1 |
| \text{June 2017} | 68.7 | b |
| \text{September 2017} | 93.5 | 1.08 |
| \text{December 2017} | 88.1 | 1.02 |
| \text{March 2018} | 91.3 | 1.06 |
| \text{June 2018} | c | 0.74 |
| \text{September 2018} | 87.5 | 1.02 |
| \text{December 2018} | 81.5 | 0.95 |
The following tables display the number of complaints for neighbourhood disturbances received by the Springfield council during weekdays for 3 weeks:
| Week | Day | Complaints | Proportion of weekly mean |
|---|---|---|---|
| 1 | \text{Monday} | 1656 | 1.47 |
| 1 | \text{Tuesday} | 254 | a |
| 1 | \text{Wednesday} | 1705 | 1.51 |
| 1 | \text{Thursday} | 878 | 0.78 |
| 1 | \text{Friday} | 1155 | 1.02 |
| 2 | \text{Monday} | 1979 | 1.52 |
| 2 | \text{Tuesday} | c | 0.54 |
| 2 | \text{Wednesday} | 1774 | 1.36 |
| 2 | \text{Thursday} | 794 | 0.61 |
| 2 | \text{Friday} | 1259 | 0.97 |
| 3 | \text{Monday} | 1433 | 0.9 |
| 3 | \text{Tuesday} | 917 | 0.57 |
| 3 | \text{Wednesday} | 1609 | 1.01 |
| 3 | \text{Thursday} | 1413 | b |
| 3 | \text{Friday} | 2609 | 1.63 |
Calculate the value of a, to two decimal places.
Calculate the value of b, to two decimal places.
If the mean for Week 2 is 1302.20, calculate the value of c, to two decimal places.
Calculate the seasonal index for Wednesday.
Seasonally adjust the data for Thursday in Week 2.
To provide information to traffic engineers, the number of vehicles using an intersection each day between 6 am and 9:30 am was recorded over three weeks. The data is tabulated below:
| \text{Week} | \text{Day} | \text{Complaints} | \text{Percentage of weeklymean} |
|---|---|---|---|
| 1 | \text{Monday} | 836 | 77\% |
| 1 | \text{Tuesday} | 1898 | 175\% |
| 1 | \text{Wednesday} | 1648 | a |
| 1 | \text{Thursday} | 1339 | 124\% |
| 1 | \text{Friday} | 640 | 58\% |
| 1 | \text{Saturday} | 980 | 90\% |
| 1 | \text{Sunday} | 241 | 22\% |
| 2 | \text{Monday} | 1321 | 94\% |
| 2 | \text{Tuesday} | 1989 | 142\% |
| 2 | \text{Wednesday} | b | 129\% |
| 2 | \text{Thursday} | 2064 | 147\% |
| 2 | \text{Friday} | 648 | 46\% |
| 2 | \text{Saturday} | 1301 | 93\% |
| 2 | \text{Sunday} | 692 | 49\% |
| 3 | \text{Monday} | 1503 | 85\% |
| 3 | \text{Tuesday} | 2635 | 148\% |
| 3 | \text{Wednesday} | 2532 | 143\% |
| 3 | \text{Thursday} | 2292 | c |
| 3 | \text{Friday} | 1136 | 64\% |
| 3 | \text{Saturday} | 1738 | 98\% |
| 3 | \text{Sunday} | 586 | 33\% |
Calculate the values of the following to the nearest percentage:
a
c
If the mean number of complaints for Week 2 is 1402.57, calculate the value of b to the nearest whole number.
Calculate the seasonal index for Monday. Round your answer to two decimal places.