A least squares regression line is fitted to some seasonally adjusted data, given by:y=3.5t+22.8
Use the regression line to predict the deseasonalised value for time period 20.
A least squares regression line is fitted to some seasonally adjusted data for time periods 1 to 15, given by:y=1.3t+45.7
Use the regression line to predict the deseasonalised value for time period 16.
Use the regression line to predict the deseasonalised value for time period 60.
Which prediction is more reliable?
The monthly average cost of a hotel room in Sydney in 2000 is shown in the following table:
| \text{Month, } t | \text{Jan} | \text{Feb} | \text{March} | \text{April} | \text{May} | \text{Jun} | \text{Jul} | \text{Aug} | \text{Sep} |
|---|---|---|---|---|---|---|---|---|---|
| \text{Hotel price }, P(\$) | 250 | 240 | 235 | 237 | 239 | 230 | 228 | 237 | 332 |
Let January 2000 be t=1 and construct a time series graph of the data.
Which month appears to be an outlier?
Remove the outlier and find the equation of the least squares regression line for the remaining data. Round all values to four decimal places.
Predict the average cost of a hotel room in Sydney in November 2000.
Data following a 5 point cyclical pattern is collected and seasonally adjusted for time periods 1 to 14. A least squares regression line is fitted to the seasonally adjusted data, which appears linear, and is given by:
y = 2.4378 t + 66.2925
Calculate the predicted deseasonalised value for time period 15, to four decimal places.
If the seasonal index for this period was 77\%, calculate the true predicted value to four decimal places.
Data following a 3 point cyclical pattern is collected and seasonally adjusted for time periods 1 to 12. A least squares regression line is fitted to the seasonally adjusted data and is given by:
y = - 2.1404 t + 51.4172
Calculate the predicted deseasonalised value for time period 16, to four decimal places.
If the seasonal index for this period was 140\%, calculate the true predicted value to four decimal places.
Comment on the reliability of the predicted value from part (b).
What does the coefficient of t indicate in the equation of the least squares regression line?
The petrol price cycle at a local service station is monitored. The results over two weeks are given in the table below:
| \text{Day} | \text{Time }(t) | \text{Price (cents)} | \text{Deseasonalised data} | |
|---|---|---|---|---|
| Week 1 | \text{Mon} | 1 | 99 | 102.97 |
| \text{Tue} | 2 | 85.2 | 98.19 | |
| \text{Wed} | 3 | 84 | 105.13 | |
| \text{Thu} | 4 | 104.7 | 103.10 | |
| \text{Fri} | 5 | 132.5 | X | |
| \text{Sat} | 6 | 113.9 | 103.15 | |
| \text{Sun} | 7 | 105.4 | 103.44 | |
| Week 2 | \text{Mon} | 8 | 114.5 | 119.10 |
| \text{Tue} | 9 | 108.1 | 124.58 | |
| \text{Wed} | 10 | 93.2 | 116.65 | |
| \text{Thu} | 11 | 120.8 | 118.96 | |
| \text{Fri} | 12 | 140.6 | 114.00 | |
| \text{Sat} | 13 | Y | 118.91 | |
| \text{Sun} | 14 | 120.8 | 118.56 |
Seasonal indices:
| Mon | Tues | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|
| 0.9614 | 0.8677 | 0.7990 | 1.0155 | 1.2333 | 1.1042 | 1.0189 |
Which is the best day of the cycle to purchase petrol?
Calculate the missing values X and Y to two decimal places.
Using your calculator, determine the equation of least squares regression line for the deseasonalised data, in terms of t. Round all values to two decimal places.
Predict the price of petrol for Thursday in the third week.
Comment on the reliability of your prediction.
A new pop up ice-cream shop records their sales over their first month. The data is tabulated below. The shop is only open from Friday to Sunday.
| \text{Day} | \text{Time }(t) | \text{Sales (dollars)} | \text{Deseasonalised data} | |
|---|---|---|---|---|
| Week 1 | \text{Fri} | 1 | 2036 | 2101.14 |
| \text{Sat} | 2 | 2257 | 2040.87 | |
| \text{Sun} | 3 | 1936 | 2092.75 | |
| Week 2 | \text{Fri} | 4 | 2224 | X |
| \text{Sat} | 5 | 2547 | 2303.10 | |
| \text{Sun} | 6 | 2060 | 2226.79 | |
| Week 3 | \text{Fri} | 7 | 2349 | 2424.15 |
| \text{Sat} | 8 | 2706 | 2446.88 | |
| \text{Sun} | 9 | Y | 2431.09 | |
| Week 4 | \text{Fri} | 10 | 2435 | 2512.90 |
| \text{Sat} | 11 | 2824 | 2553.58 | |
| \text{Sun} | 12 | 2398 | 2592.15 |
Seasonal indices:
| Fri | Sat | Sun |
|---|---|---|
| 0.9690 | 1.1059 | 0.9251 |
On which day will shop be most likely to need extra help?
Calculate the missing values X and Y to two decimal places.
Using your calculator, determine the equation of least squares regression line for the deseasonalised data in terms of t. Round all values to two decimal places.
Predict the sales for Friday of the sixth week.
Comment on the reliability of your prediction
The number of customers served at a shopping centre cafe are recorded quarterly over a period of four years and the results are entered into the table below:
| \text{Month} | \text{Time }(t) | \text{Customer} \\ \text{numbers} | \text{Cycle} \\ \text{mean} | \text{Perentage} \\ \text{of cycle mean} | \text{Deaseasonalised} \\ \text{number} | |
|---|---|---|---|---|---|---|
| 2016 | \text{Jan} | 1 | 1687 | 130.396\% | 1284 | |
| \text{Apr} | 2 | 1218 | 94.145\% | 1256 | ||
| \text{Jul} | 3 | 886 | 1293.75 | 68.483\% | 1329 | |
| \text{Oct} | 4 | 1384 | 106.976\% | 1318 | ||
| 2017 | \text{Jan} | 5 | 1789 | 130.823\% | 1361 | |
| \text{Apr} | 6 | 1327 | 97.038\% | 1369 | ||
| \text{Jul} | 7 | 905 | 1367.5 | 66.179\% | 1358 | |
| \text{Oct} | 8 | 1449 | 105.960\% | 1380 | ||
| 2018 | \text{Jan} | 9 | 2325 | 132.971\% | 1769 | |
| \text{Apr} | 10 | 1745 | 99.800\% | 1800 | ||
| \text{Jul} | 11 | 1112 | 1748.5 | 63.597\% | 1669 | |
| \text{Oct} | 12 | 1821 | 103.632\% | 1726 | ||
| 2019 | \text{Jan} | 13 | 2565 | 131.471\% | 1952 | |
| \text{Apr} | 14 | 1890 | 96.873\% | 1949 | ||
| \text{Jul} | 15 | 1333 | 1951 | 68.324\% | 2000 | |
| \text{Oct} | 16 | 2016 | 103.332\% | 1920 |
Seasonal indices:
| Jan | Apr | Jul | Oct |
|---|---|---|---|
| 131.42\% | 96.964\% | 66.646\% | 104.975\% |
Use your calculator to determine the equation of the least squares regression line for the deseasonalised data, in terms of t. Round all values to three decimal places.
What does the coefficient of t indicate in the least squares regression line?
State the value of t for April 2020.
Use the regression line from the deseasonalised data and the seasonal index for April to predict the number of customers for April 2020. Round your answer to the nearest whole number.
Comment on the reliability of your prediction.
The cafe owner used the following calculation to predict the number of customers for July 2021:
\begin{aligned} \text{Predicted Value} & = \left( 55.1588 \times 19 + 1121.15\right) \times \dfrac{66.464}{100} \\ & = 1141.71\ldots \\ & \approx 1142 \end{aligned}What is wrong with this prediction?
The number of customers served at a shopping centre cafe are recorded quarterly over a period of four years and the results are entered into the table below:
| \text{Month} | \text{Time }(t) | \text{Customer numbers} | \text{4CMA} | |
|---|---|---|---|---|
| 2016 | \text{Jan} | 1 | 1687 | |
| \text{Apr} | 2 | 1218 | ||
| \text{Jul} | 3 | 886 | 1306.50 | |
| \text{Oct} | 4 | 1384 | 1332.88 | |
| 2017 | \text{Jan} | 5 | 1789 | 1348.88 |
| \text{Apr} | 6 | 1327 | 1359.38 | |
| \text{Jul} | 7 | 905 | 1434.50 | |
| \text{Oct} | 8 | 1449 | 1553.75 | |
| 2018 | \text{Jan} | 9 | 2325 | 1631.88 |
| \text{Apr} | 10 | 1745 | 1703.13 | |
| \text{Jul} | 11 | 1112 | 1778.50 | |
| \text{Oct} | 12 | 1812 | 1826.63 | |
| 2019 | \text{Jan} | 13 | 2565 | 1872.38 |
| \text{Apr} | 14 | 1890 | 1925.50 | |
| \text{Jul} | 15 | 1333 | ||
| \text{Oct} | 16 | 2016 |
Seasonal indices:
| Jan | Apr | Jul | Oct |
|---|---|---|---|
| 131.42\% | 96.964\% | 66.646\% | 104.975\% |
Use your calculator to determine the equation of the least squares regression line for the 4CMA data, in terms of t. Round all values to four decimal places.
What does the coefficient of t indicate in the least squares regression line?
State the value of t for January 2021.
Use the equation of the regression line from the 4CMA data and the seasonal index for January to predict the number of customers for January 2021. Round your answer to the nearest whole number.
Comment on the reliability of your prediction.
The cafe owner used the following calculation to predict the number of customers for October 2020.
\begin{aligned} \text{Predicted Value} & = 62.8964 \times 20 + 1054.8731 \\ & = 3122.14 \ldots \\ & \approx 2313 \end{aligned}What is wrong with the prediction?
The number of members attending a gym are recorded weekly over a period of four months and the results are entered into the table below. The owner decides to use a 4 point centred moving average to smooth the data and make predictions.
| \text{Week} | \text{Time }(t) | \text{Attendance numbers} | \text{4CMA} | |
|---|---|---|---|---|
| Jan | 1 | 1 | 760 | |
| 2 | 2 | 1123 | ||
| 3 | 3 | 815 | 1073.25 | |
| 4 | 4 | 1560 | 1100.00 | |
| Feb | 1 | 5 | 830 | 1129.00 |
| 2 | 6 | 1267 | 1156.25 | |
| 3 | 7 | 903 | 1173.13 | |
| 4 | 8 | 1690 | 1188.75 | |
| Mar | 1 | 9 | 835 | 1210.13 |
| 2 | 10 | 1387 | 1219.25 | |
| 3 | 11 | 954 | 1227.63 | |
| 4 | 12 | 1712 | 1249.88 | |
| Apr | 1 | 13 | 880 | 1274.75 |
| 2 | 14 | 1520 | 1293.88 | |
| 3 | 15 | 1020 | ||
| 4 | 16 | 1799 |
Seasonal indices:
| Week 1 | Week 2 | Week 3 | Week 4 |
|---|---|---|---|
| 69.49\% | 110.889\% | 77.455\% | 142.166\% |
Use your calculator to determine the equation of the least squares regression line for the 4CMA data, in terms of t. Round all values to four decimal places.
What does the coefficient of t indicate in the least squares regression line?
State the value of t for Week 2 of May.
Use the equation of the regression line from the 4CMA data and the seasonal index for Week 2 to predict the number of customers for Week 2 May. Round your answer to the nearest whole number.
Comment on the reliability of your prediction.
The gym owner used the following calculation to predict the attendance for Week 2 June.
\begin{aligned} \text{Predicted Value} & = \left( 18.7499 \times 22 + 1031.9506\right) \times 110.889 \\ & = 160\,173.44 \ldots \\ & \approx 160\,173 \end{aligned}What is wrong with this prediction?
The number of hockey sticks sold are recorded tri-annually over a period of four years and the results are entered into the table below. The owner decides to use a 3 point centred moving average to smooth the data and make predictions.
Use your calculator to determine the equation of the least squares regression line for the 3MA data, in terms of t. Round all values to two decimal places.
What does the coefficient of t indicate in the least squares regression line?
State the value of t for May 2021.
Predict the whole number of hockey sticks sold in May 2021.
Comment on the reliability of your prediction.
The sports store owner used the following calculation to predict the sales for May 2022:
\begin{aligned} \left( 1.16 \times 14 + 31.06\right) \times \frac{100}{171.56} & = 27.57 \ldots \\ & \approx 28 \end{aligned}What is wrong with the prediction?
| \text{Month} | \text{Time } \\\ (t) | \text{Sales} | \text{3MA} | |
|---|---|---|---|---|
| 2016 | \text{Jan} | 1 | 12 | |
| \text{May} | 2 | 54 | 31.33 | |
| \text{Sep} | 3 | 25 | 34.00 | |
| 2017 | \text{Jan} | 4 | 15 | 36.33 |
| \text{May} | 5 | 62 | 36.67 | |
| \text{Sep} | 6 | 32 | 39.00 | |
| 2018 | \text{Jan} | 7 | 16 | 40.67 |
| \text{May} | 8 | 69 | 41.33 | |
| \text{Sep} | 9 | 37 | 43.33 | |
| 2019 | \text{Jan} | 10 | 18 | 44.67 |
| \text{May} | 11 | 75 | 38.67 | |
| \text{Sep} | 12 | 41 |
Seasonal indices:
| Jan | May | Sep |
|---|---|---|
| 40.12\% | 171.56\% | 88.32\% |
The number of students cycling to the university library is recorded daily over a period of three weeks and the results are entered into the table to the right. A 7 point moving average is used to smooth the data in order to make predictions. The seasonal indices have been calculated in the table below using the average percentage method.
Use your calculator to determine the equation of the least squares regression line for the 7MA data, in terms of t. Round all values to four decimal places.
What does the coefficient of t indicate in the least squares regression line?
State the value of t for Saturday Week 4.
Predict the number of cyclists for Saturday Week 4. Round your answer to the nearest whole number.
Comment on the reliability of your prediction.
| \text{Week} | \text{Day} | \text{Time } \\ (t) | \text{No.} | \text{7MA} |
|---|---|---|---|---|
| 1 | \text{Mon} | 1 | 66 | |
| \text{Tues} | 2 | 68 | ||
| \text{Wed} | 3 | 71 | ||
| \text{Thu} | 4 | 65 | 103.57 | |
| \text{Fri} | 5 | 255 | 103.86 | |
| \text{Sat} | 6 | 111 | 103.71 | |
| \text{Sun} | 7 | 89 | 103.86 | |
| 2 | \text{Mon} | 8 | 68 | 104.00 |
| \text{Tues} | 9 | 67 | 103.29 | |
| \text{Wed} | 10 | 72 | 102.71 | |
| \text{Thu} | 11 | 66 | 102.14 | |
| \text{Fri} | 12 | 250 | 101.86 | |
| \text{Sat} | 13 | 107 | 102.00 | |
| \text{Sun} | 14 | 85 | 102.00 | |
| 3 | \text{Mon} | 15 | 66 | 101.43 |
| \text{Tues} | 16 | 68 | 100.57 | |
| \text{Wed} | 17 | 72 | 99.43 | |
| \text{Thu} | 18 | 62 | 98.57 | |
| \text{Fri} | 19 | 244 | ||
| \text{Sat} | 20 | 99 | ||
| \text{Sun} | 21 | 79 |
Seasonal indices:
| Mon | Tues | Wed | Thu | Fri | Sat | Sun |
|---|---|---|---|---|---|---|
| 0.66 | 0.67 | 0.71 | 0.63 | 2.46 | 1.04 | 0.83 |
The following data shows the sales of washing machines at a leading retailer over four quarters of three consecutive years:
| \text{Month} | \text{Time }(t) | \text{Number of}\\\text{ washing machines sold} | \text{Percentage of}\\\text{yearly mean} | \text{4CMA} | |
|---|---|---|---|---|---|
| Year 1 | \text{March} | 1 | 455 | 39.014\% | |
| \text{June} | 2 | 1054 | 90.375\% | ||
| \text{Sept} | 3 | 613 | 52.562\% | 1167.63 | |
| \text{Dec} | 4 | 2543 | 218.049\% | 1113.38 | |
| Year 2 | \text{March} | 5 | 466 | 40.303\% | 1063.00 |
| \text{June} | 6 | 609 | 52.670\% | X | |
| \text{Sept} | 7 | 655 | 56.649\% | 1168.63 | |
| \text{Dec} | 8 | 2895 | 250.378\% | 1252.50 | |
| Year 3 | \text{March} | 9 | 565 | 38.176\% | 1328.00 |
| \text{June} | 10 | 1181 | 79.797\% | Y | |
| \text{Sept} | 11 | 687 | 46.419\% | ||
| \text{Dec} | 12 | 3487 | 235.608\% |
Calculate the seasonal index, correct to three decimal places for the quarters ending in:
March
June
September
December
The data is smoothed using a 4 point centred moving average as shown in the table above. Calculate the missing values X and Y.
Use your calculator to determine the equation of the least squares regression line for the 4CMA data, in terms of t. Round all values to four decimal places.
Predict the number of washing machines sold in the quarter ending September Year 5. Give your answer to the nearest whole number.
Comment on the reliability of your prediction.
The following data shows the number of customers at a hand car wash business for the first 4 weeks of 3 consecutive months:
| \text{Week} | \text{Time }(t) | \text{Number of}\\\text{ customers} | \text{Percentage of}\\\text{ monthly mean} | \text{Deseasonalised}\\\text{ data} | |
|---|---|---|---|---|---|
| March | \text{1} | 1 | 800 | 112.44\% | A |
| \text{2} | 2 | 743 | 104.43\% | 726.92 | |
| \text{3} | 3 | 453 | 63.67\% | 722.30 | |
| \text{4} | 4 | 850 | 119.47\% | 705.88 | |
| April | \text{1} | 5 | 780 | 116.46\% | 680.31 |
| \text{2} | 6 | 676 | 100.93\% | B | |
| \text{3} | 7 | 423 | 63.16\% | 674.46 | |
| \text{4} | 8 | 800 | 119.45\% | 664.36 | |
| May | \text{1} | 9 | 743 | 115.06\% | 648.04 |
| \text{2} | 10 | 654 | 101.28\% | 639.84 | |
| \text{3} | 11 | 396 | 61.23\% | C | |
| \text{4} | 12 | 790 | 122.34\% | 656.05 |
Calculate the seasonal index, correct to five decimal places for each of the following weeks:
Week 1
Week 2
Week 3
Week 4
The data is smoothed by deseasonalising the data as shown in the table above. Calculate the missing values A, B and C correct to two decimal places.
Use your calculator to determine the equation of the least squares regression line for the deseasonalised data, in terms of t. Round all values to four decimal places.
Predict the number of customers for Week 4 in June. Round your answer to the nearest whole number.
Comment on the reliability of your prediction.
The following data shows the sales of air conditioners at a leading retailer over four quarters from 2017 to 2019:
| \text{Month} | \text{Time }(t) | \text{No. of air}\\ \text{conditioners sold} | \text{Proportion}\\ \text{of yearly mean} | \text{Deseasonalised}\\ \text{data} | |
|---|---|---|---|---|---|
| 2017 | \text{March} | 1 | 332 | 0.3054 | |
| \text{June} | 2 | 320 | 0.2943 | ||
| \text{Sept} | 3 | 966 | 0.8885 | ||
| \text{Dec} | 4 | 2731 | 2.5118 | ||
| 2018 | \text{March} | 5 | 987 | 0.6340 | |
| \text{June} | 6 | 926 | 0.5948 | ||
| \text{Sept} | 7 | 1117 | 0.7175 | ||
| \text{Dec} | 8 | 3197 | 2.0536 | ||
| 2019 | \text{March} | 9 | 1216 | 0.6910 | |
| \text{June} | 10 | 939 | 0.5336 | ||
| \text{Sept} | 11 | 1414 | 0.8035 | ||
| \text{Dec} | 12 | 3470 | 1.9719 |
Deseasonalise the data and use a least squares regression line to predict the whole number of air conditioners sold in the quarter ending December 2020.
The following data shows the sales of air conditioners at a leading retailer over four quarters of three consecutive years:
| \text{Month} | \text{Time }(t) | \text{Number of}\\ \text{ air conditioners sold} | \text{Proportion}\\ \text{ of yearly mean} | \text{4CMA} | |
|---|---|---|---|---|---|
| Year 1 | \text{March} | 1 | 1042 | 0.8529 | |
| \text{June} | 2 | 486 | 0.3978 | ||
| \text{Sept} | 3 | 613 | 0.5017 | 1236.5 | |
| \text{Dec} | 4 | 2746 | 2.2476 | 1266.625 | |
| Year 2 | \text{March} | 5 | 1160 | 0.8183 | 1347.75 |
| \text{June} | 6 | 609 | 0.4296 | A | |
| \text{Sept} | 7 | 1139 | 0.8035 | 1496.875 | |
| \text{Dec} | 8 | 2762 | 1.9485 | 1647.75 | |
| Year 3 | \text{March} | 9 | 1795 | 0.9638 | 1713.625 |
| \text{June} | 10 | 1181 | 0.6341 | B | |
| \text{Sept} | 11 | 1094 | 0.5874 | ||
| \text{Dec} | 12 | 3380 | 1.8148 |
Calculate the seasonal index, correct to four decimal places for the quarters ending in:
March
June
September
December
The data is smoothed using a 4 point centred moving average as shown in the table below. Calculate the missing values A and B.
Use your calculator to calculate the equation of the least squares regression line that fits the 4CMA data, in terms of t. Round all values to four decimal places.
Predict the whole number of air conditioners sold in the quarter ending December Year 4.
Comment on the reliability of your prediction.
The electricity bills of an energy conscious household are noted over two years. The data is represented in the table below:
| \text{Month} | \text{Time }(t) | \text{Billed amount} | \text{Proportion}\\ \text{of yearly mean} | \text{Deseasonalised}\\ \text{data} | |
|---|---|---|---|---|---|
| 2018 | \text{Feb} | 1 | 122 | 0.7114 | |
| \text{Apr} | 2 | 242 | 1.4111 | ||
| \text{Jun} | 3 | 110 | 0.6414 | ||
| \text{Aug} | 4 | 156 | 0.9096 | ||
| \text{Oct} | 5 | 159 | 0.9271 | ||
| \text{Dec} | 6 | 240 | 1.3994 | ||
| 2019 | \text{Feb} | 7 | 66 | 0.5287 | |
| \text{Apr} | 8 | 169 | 1.3538 | ||
| \text{Jun} | 9 | 84 | 0.6729 | ||
| \text{Aug} | 10 | 127 | 1.0174 | ||
| \text{Oct} | 11 | 124 | 0.9933 | ||
| \text{Dec} | 12 | 179 | 1.4339 |
Deseasonalise the data in the table and complete the last column and find the least squares regression line that fits deseasonalised data in terms of t, and use the model to predict the electricity bill amount for February 2021.
A nursery records the demand of herb seedlings by making note of the sales during periods of the year. The results are given below:
| t | \text{Time period} | \text{Seedlings sold } | \text{Deseasonalised data} |
|---|---|---|---|
| 1 | \text{Apr 2017} | 1636 | 1959.05 |
| 2 | \text{Aug 2017} | 1472 | 1957.19 |
| 3 | \text{Dec 2017} | 2977 | 2107.16 |
| 4 | \text{Apr 2018} | 2027 | 2427.25 |
| 5 | \text{Aug 2018} | 1730 | 2300.23 |
| 6 | \text{Dec 2018} | 3463 | 2451.16 |
| 7 | \text{Apr 2019} | 2342 | 2804.45 |
| 8 | \text{Aug 2019} | 2219 | 2950.41 |
| 9 | \text{Dec 2019} | 3641 | 2577.15 |
Seasonal indices:
| April | August | December |
|---|---|---|
| 0.8351 | 0.7521 | 1.4128 |
Use a least squares regression line to predict the whole number of seedlings sold in August 2020.