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.
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:
Which day tends to be the busiest for sausage sellers?
What type of moving average would best smooth the data?
| Day | Sales | |
|---|---|---|
| Week 3 | \text{Fri} | 162 |
| \text{Sat} | 484 | |
| \text{Sun} | 600 | |
| Week 4 | \text{Fri} | 195 |
| \text{Sat} | 491 | |
| \text{Sun} | 677 |
The following table shows some time series data where t represents time:
Calculate the 5 point moving average at
t = 3.
Calculate the 4 point centred moving average at t = 3.
| t | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| y | 12 | 14 | 22 | 11 | 16 | 25 |
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) | Mon | Tue | Wed | Thu | Fri |
|---|---|---|---|---|---|
| \text{Number of children} | 75 | 150 | 162 | 205 | 243 |
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.
The seasonal indices are shown in the table below. Complete the table by finding the seasonal index for Wednesday.
| Day (in Week 4) | Mon | Tue | Wed | Thu | Fri |
|---|---|---|---|---|---|
| \text{Seasonal indices} | 40.22\% | 78.09\% | 134.05\% | 162.60\% |
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.
The deseasonalised number of visitors for Thursday Week 3 is 110.4. Determine the actual number of visitors for this day.
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:
Calculate the 3 point moving average for Sunday of Week 3. Round your answer to two decimal places.
| Day | Sales | |
|---|---|---|
| Week 3 | \text{Fri} | 145 |
| \text{Sat} | 492 | |
| \text{Sun} | 604 | |
| Week 4 | \text{Fri} | 180 |
| \text{Sat} | 530 | |
| \text{Sun} | 675 |
The seasonal indices for Friday and Sunday are shown in the second table. Calculate the seasonal index for Saturday.
| Friday | Saturday | Sunday |
|---|---|---|
| 36.11\% | 143.70\% |
Complete the third table by calculating the deseasonalised figures of Week 4. Round your answers to the nearest whole number.
| Day | Raw data | Deseasonalised data |
|---|---|---|
| \text{Fri} | 180 | |
| \text{Sat} | 530 | |
| \text{Sun} | 675 |
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:
| Day | Sales | |
|---|---|---|
| Week 3 | \text{Fri} | 156 |
| \text{Sat} | 487 | |
| \text{Sun} | 586 | |
| Week 4 | \text{Fri} | 204 |
| \text{Sat} | 485 | |
| \text{Sun} | 645 |
| Fri | Sat | Sun | |
|---|---|---|---|
| Seasonal index | 42.85\% | 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.
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} | 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 |
Seasonal indices:
| Trimester | Jan | May | Sept |
|---|---|---|---|
| \text{Seasonal index} | E | 0.9545 | 0.7245 |
Determine the value A.
Determine the value B, to two decimal places.
Determine the value C as a percentage rounded to one decimal place.
Given the seasonal index for May, determine the value D.
Find E, the seasonal index for January.
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.
Comment on the reliability of your prediction.
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} | 1 | 1087 | 1223.5 | 88.8\% | |
| \text{Sat (matinee)} | 2 | 844 | 69.0\% | |||
| \text{Sat (evening)} | 3 | 1731 | 1231.0 | 141.5\% | ||
| \text{Sun} | 4 | 1232 | Y | 100.7\% | ||
| Week 2 | \text{Fri} | 5 | 1003 | 1192.0 | 1153.5 | 87.0\% |
| \text{Sat (matinee)} | 6 | 802 | 1172.8 | 69.5\% | ||
| \text{Sat (evening)} | 7 | 1731 | 1147.3 | 150.1\% | ||
| \text{Sun} | 8 | 1078 | 1133.6 | 93.5\% | ||
| Week 3 | \text{Fri} | 9 | 953 | 1121.1 | 1085 | 87.8\% |
| \text{Sat (matinee)} | 10 | 743 | 1100.5 | 68.5\% | ||
| \text{Sat (evening)} | 11 | 1690 | 1078.3 | 155.8\% | ||
| \text{Sun} | 12 | 954 | 1059.9 | 87.9\% | ||
| Week 4 | \text{Fri} | 13 | 899 | 1034.5 | 998.5 | 90.0\% |
| \text{Sat (matinee)} | 14 | 650 | 1009.6 | 65.1\% | ||
| \text{Sat (evening)} | 15 | 1580 | 158.2\% | |||
| \text{Sun} | 16 | X | 86.6\% |
What is the break even attendance figure for the ballet company? Round your answer to the nearest whole number.
Find the value of:
X
Y
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.
Use the average percentage method to calculate the seasonal index for the Friday performances.
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.
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.
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.
Comment on the reliability of your prediction.
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} | 0.9 | 2.1 | 42.86\% | D |
| 2 | \text{Aug} | 2.4 | 114.29\% | 2.2 | ||
| 3 | \text{Dec} | 3.0 | C | 2.1 | ||
| 2018 | 4 | \text{Apr} | 1.4 | 2.8 | 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} | 2.2 | B | 42.31\% | 4.9 |
| 8 | \text{Aug} | 6.0 | 115.38\% | 5.2 | ||
| 9 | \text{Dec} | 7.3 | 140.38\% | 5.2 |
Determine the values of the following to one decimal place:
A
B
D
Determine the value C as a percentage rounded to two decimal places.
Calculate the seasonal index for April. Give your answer as a percentage and round your answer to three decimal places.
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).
Which method of smoothing seems to be the most appropriate for this situation?
Which type of moving average should Emma use to smooth the data?
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} | 2.4 | 2.10 | |
| 3 | \text{Dec} | 3.0 | 2.27 | |
| 2018 | 4 | \text{Apr} | 1.4 | B |
| 5 | \text{Aug} | 3.2 | 2.80 | |
| 6 | \text{Dec} | 3.8 | 2.97 | |
| 2019 | 7 | \text{Apr} | A | 3.90 |
| 8 | \text{Aug} | 6.0 | 5.07 | |
| 9 | \text{Dec} | 7.3 |
Determine the value of:
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?
Comment on the reliability of Emma's prediction.
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} | 1 | 134 | 154.2 | 86.9\% |
| \text{Tues} | 2 | 154 | 99.87\% | ||
| \text{Wed} | 3 | 143 | 92.74\% | ||
| \text{Thurs} | 4 | 165 | 107.00\% | ||
| \text{Fri} | 5 | 175 | 113.49\% | ||
| Week 2 | \text{Mon} | 6 | 137 | 157.6 | 86.93\% |
| \text{Tues} | 7 | 158 | 100.25\% | ||
| \text{Wed} | 8 | 145 | 92.01\% | ||
| \text{Thurs} | 9 | 169 | 107.23\% | ||
| \text{Fri} | 10 | 179 | 113.58\% | ||
| Week 3 | \text{Mon} | 11 | 140 | 159.6 | 87.72\% |
| \text{Tues} | 12 | 160 | 100.25\% | ||
| \text{Wed} | 13 | 147 | 92.11\% | ||
| \text{Thurs} | 14 | 171 | 107.14\% | ||
| \text{Fri} | 15 | 180 | 112.78\% | ||
| Week 4 | \text{Mon} | 16 | 145 | 164.2 | 88.31\% |
| \text{Tues} | 17 | 163 | 99.27\% | ||
| \text{Wed} | 18 | 151 | 91.96\% | ||
| \text{Thurs} | 19 | 176 | 107.19\% | ||
| \text{Fri} | 20 | 186 | 113.28\% |
| Day | Monday | Tuesday | Wednesday | Thursday | Friday |
|---|---|---|---|---|---|
| \text{Seasonal index} | 0.8746 | 0.9991 | 0.9221 | 1.0714 | 1.1328 |
Use a least squares regression line to predict the actual number of coffee pods used for Tuesday of Week 6.
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} | 1 | 136 | 375.00 | 36.27\% |
| \text{Sat} | 2 | 443 | 118.13\% | ||
| \text{Sun} | 3 | 546 | 145.60\% | ||
| Week 2 | \text{Fri} | 4 | 140 | 377.33 | 37.10\% |
| \text{Sat} | 5 | A | 116.08\% | ||
| \text{Sun} | 6 | 554 | 146.82\% | ||
| Week 3 | \text{Fri} | 7 | 132 | 378.67 | C |
| \text{Sat} | 8 | 443 | 119.89\% | ||
| \text{Sun} | 9 | 550 | 145.25\% | ||
| Week 4 | \text{Fri} | 10 | 135 | B | 35.75\% |
| \text{Sat} | 11 | 441 | 116.77\% | ||
| \text{Sun} | 12 | 557 | 147.48\% |
Determine the value of:
A
B
C
Complete the given table by calculating the seasonal index for Sunday.
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.
| Day | Seasonal index |
|---|---|
| \text{Friday} | 0.3599 |
| \text{Saturday} | 1.1772 |
| \text{Sunday} |
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.
Predict the actual number of sausages sold on Saturday of Week 5.
Comment on the reliability of your prediction.
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} | 1 | 21 | 39.47\% | 41.6 | |
| \text{Tues} | 2 | 44 | 82.71\% | 41.6 | ||
| \text{Wed} | 3 | 48 | 53.2 | 90.23\% | 51.2 | |
| \text{Thurs} | 4 | A | 122.18\% | 57.8 | ||
| \text{Fri} | 5 | 88 | 165.41\% | 56.5 | ||
| Week 2 | \text{Mon} | 6 | 33 | 52.72\% | 65.4 | |
| \text{Tues} | 7 | 57 | 91.05\% | 65.1 | ||
| \text{Wed} | 8 | 61 | B | 97.44\% | 65.1 | |
| \text{Thurs} | 9 | 70 | 111.82\% | 62.3 | ||
| \text{Fri} | 10 | 92 | 146.96\% | 59.1 | ||
| Week 3 | \text{Mon} | 11 | 38 | 53.37\% | 75.3 | |
| \text{Tues} | 12 | 63 | 88.48\% | 71.9 | ||
| \text{Wed} | 13 | 66 | 71.2 | C | 70.4 | |
| \text{Thurs} | 14 | 77 | 108.15\% | 68.5 | ||
| \text{Fri} | 15 | 112 | 157.30\% | 71.9 | ||
| Week 4 | \text{Mon} | 16 | 44 | 56.27\% | 87.2 | |
| \text{Tues} | 17 | 69 | 88.24\% | D | ||
| \text{Wed} | 18 | 74 | 78.2 | 95.63\% | 78.9 | |
| \text{Thurs} | 19 | 84 | 107.42\% | 74.7 | ||
| \text{Fri} | 20 | 120 | 153.45\% | 77.0 |
Determine the values of the following. Round your answers to one decimal place if necessary.
A
B
D
Determine the value C as a percentage rounded to two decimal places.
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.
Calculate the seasonal index for Friday. Give your answer as a percentage.
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.
Comment on the reliability of your prediction.
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:
Complete the time series plot by including this additional information.
What type of moving average would best smooth this data?
Explain why this moving average is the best fit for this data.
| No. of Children | |
|---|---|
| Wk 4 - Mon | 75 |
| Wk 4 - Tue | 150 |
| Wk 4 - Wed | 162 |
| Wk 4 - Thu | 174 |
| Wk 4 - Fri | 180 |
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:
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.
Graph this line and the above data on the same number plane.
Describe on the overall trend of the data.