The following graph displays yearly average interest rates plotted over a period of twelve years. A trend line has been fitted to the data as shown.
Calculate the equation of the trend line.
A least squares regression line is fitted to seasonally adjusted data.
For time period t = 5, the predicted deseasonalised value is 44.9036.
For time period t = 10, the predicted deseasonalised value is 60.3796.
Determine the equation of the least squares regression line in the form y = a t + b.
The number of rainy days per month is recorded at a weather station. In the following table, the number of rainy days per month is plotted for January (Month 1) to August (Month 8) in the same year:
| \text{Month number}, x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| \text{Number of rainy days}, y | 11 | 9 | 11 | 15 | 18 | 17 | 21 | 19 |
Determine the equation of the least squares line for this data, rounding all values to one decimal place.
Use your CAS calculator to plot the least squares line and the time series plot on the same set of axes.
The forecast equation for calculating the share price, y, of a bank is obtained from the data of the bank's share price over the past seven years.
The equation is y = 22.74 + 0.28 t, where t = 1 represents the Year 2007.
Which of the following corresponds to the equation in the context of the question?
\text{year} = -1983.26+0.28\times\text{share price}
\text{share price} = 0.28+22.47\times\text{year}
\text{share price} = -538.94+0.28\times\text{year}
\text{share price} = -1983.26+0.28\times\text{year}
\text{year} = 22.74+0.28\times\text{share price}
Predict the share price in 2017.
The least squares regression line y = 4.5493 t + 48.9941 is fitted to seasonally adjusted data.
Calculate the predicted deseasonalised value for time period 15. Round your answers to four decimal places.
If the seasonal index for this period was 69\%, calculate the true predicted value. Round your answers to four decimal places.
The least squares regression line y =- 2.7698 t + 79.3735 is fitted to seasonally adjusted data.
Calculate the predicted deseasonalised value for time period 16. Round your answers to four decimal places.
If the seasonal index for this period was 116\%, calculate the true predicted value. Round your answers to four decimal places.
The following table displays the percentage of total retail sales that were made in department stores over an eleven-year period:
Use your CAS calculator to construct a time series plot for this data.
Describe the trend of the time series plot.
Find the equation of the trend line.
Explain the meaning of the slope of the trend line.
Use the trend line to forecast the percentage of retail sales which will be made by department stores in Year 15. Round your answer to one decimal place.
| \text{Year} | \text{Sales}(\%) |
|---|---|
| 1 | 12.3 |
| 2 | 12.0 |
| 3 | 11.7 |
| 4 | 11.5 |
| 5 | 11.0 |
| 6 | 10.5 |
| 7 | 10.6 |
| 8 | 10.7 |
| 9 | 10.4 |
| 10 | 10.0 |
| 11 | 9.4 |
The monthly average cost of a hotel room in Sydney in 2019 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 }(\$) | 250 | 240 | 235 | 237 | 239 | 230 | 228 | 237 | 332 |
Use your CAS calculator to plot the data. Let January 2019 be t=1.
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 the numbers to four decimal places.
Predict the average cost of a hotel room in Sydney in November 2019.
A nursery records the demand of herb seedlings by making note of the sales during three time periods each year. The results and seasonal components are displayed in the given tables:
Use your CAS calculator to find the least squares regression line that fits the deseasonalised data, where t = 1 is April 2017.
Give the equation of the line in the form y = a t + b and round any numbers to two decimal places.
Predict the number of seedlings sold in August 2020.
Comment on the reliability of your prediction.
Predict the number of seedlings sold in April 2021.
Comment on the reliability of this second prediction.
| Time period | Number of seedlings sold | Deseasonalised data |
|---|---|---|
| \text{Apr 2017} | 1636 | 1959.05 |
| \text{Aug 2017} | 1472 | 1957.19 |
| \text{Dec 2017} | 2977 | 2107.16 |
| \text{Apr 2018} | 2027 | 2427.25 |
| \text{Aug 2018} | 1730 | 2300.23 |
| \text{Dec 2018} | 3463 | 2451.16 |
| \text{Apr 2019} | 2342 | 2804.45 |
| \text{Aug 2019} | 2219 | 2950.41 |
| \text{Dec 2019} | 3641 | 2577.15 |
Seasonal Indices:
| April | August | December |
|---|---|---|
| 0.8351 | 0.7521 | 1.4128 |
The petrol price cycle at a local service station is monitored over a two week period. The results and seasonal indices are displayed in the following tables:
| Day | Price (cents) | Deseasonalised data |
|---|---|---|
| \text{Mon Week 1} | 99 | 102.97 |
| \text{Tue Week 1} | 85.2 | 98.19 |
| \text{Wed Week 1} | 84 | 105.13 |
| \text{Thu Week 1} | 104.7 | 103.10 |
| \text{Fri Week 1} | 132.5 | X |
| \text{Sat Week 1} | 113.9 | 103.15 |
| \text{Sun Week 1} | 105.4 | 103.44 |
| \text{Mon Week 2} | 114.5 | 119.10 |
| \text{Tue Week 2} | 108.1 | 124.58 |
| \text{Wed Week 2} | 93.2 | 116.65 |
| \text{Thu Week 2} | 120.8 | 118.96 |
| \text{Fri Week 2} | 140.6 | 114.00 |
| \text{Sat Week 2} | Y | 118.91 |
| \text{Sun Week 2} | 120.8 | 118.56 |
Seasonal Indices:
| Mon | Tue | 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 values of X. Round your answer to two decimal places.
Calculate the values of Y. Round your answer to one decimal place.
Using your CAS calculator, determine the equation of the least squares regression line for the deseasonalised data, where t = 1 is Monday of Week 1.
Give the equation in the form y = a t + b and round any figures 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 in whole dollars over their first month. The data and seasonal indices are in the tables below. Note that the shop is only open three days per week.
| Day | Sales ($) | Deseasonalised data |
|---|---|---|
| \text{Fri Week 1} | 2036 | 2101.14 |
| \text{Sat Week 1} | 2257 | 2040.87 |
| \text{Sun Week 1} | 1936 | 2092.75 |
| \text{Fri Week 2} | 2224 | X |
| \text{Sat Week 2} | 2547 | 2303.10 |
| \text{Sun Week 2} | 2060 | 2226.79 |
| \text{Fri Week 3} | 2349 | 2424.15 |
| \text{Sat Week 3} | 2706 | 2446.88 |
| \text{Sun Week 3} | Y | 2431.09 |
| \text{Fri Week 4} | 2435 | 2512.90 |
| \text{Sat Week 4} | 2824 | 2553.58 |
| \text{Sun Week 4} | 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 value of X in the table. Round your answer to two decimal places.
Calculate the value of Y in the table. Round your answer to the nearest whole number.
Using your CAS calculator, determine the equation of the least squares regression line for the deseasonalised data, where t = 1 is Friday of Week 1.
Give the equation in the form y = a t + b and round any numbers to two decimal places.
Predict the sales for Friday of the sixth week.
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:
| Time Period | Number of Air Conditioners Sold | Proportion of Yearly Mean | Deasonalised data |
|---|---|---|---|
| \text{1 (March 2017}) | 1042 | 0.8529 | |
| \text{2 (June 2017}) | 486 | 0.3978 | |
| \text{3 (Sept 2017}) | 613 | 0.5017 | |
| \text{4 (Dec 2017}) | 2746 | 2.2476 | |
| \text{5 (March 2018}) | 1160 | 0.8183 | |
| \text{6 (June 2018}) | 609 | 0.4296 | |
| \text{7 (Sept 2018}) | 1139 | 0.8035 | |
| \text{8 (Dec 2018}) | 2762 | 1.9485 | |
| \text{9 (March 2019}) | 1795 | 0.9638 | |
| \text{10 (June 2019}) | 1181 | 0.6341 | |
| \text{11 (Sept 2019}) | 1094 | 0.5874 | |
| \text{12 (Dec 2019}) | 3380 | 1.8148 |
Calculate the seasonal component for each quarter to four decimal places:
March
June
September
December
Deseasonalise the data in the table. Round to the nearest whole air conditioner sold.
Use your CAS calculator to calculate the least squares regression line for the deseasonalised data, rounding values to one decimal place.
Give the equation of the line in the form y = a t + b, where t=1 for March 2017.
Predict the deseasonalised number of air conditioners sold in the quarter ending December 2020. Round your answer to one decimal place.
Predict the actual number of air conditioners sold in the quarter ending December 2020.
Comment on the reliability of your prediction.
The electricity bills of an energy conscious household are noted over a two year period. The data is represented in the following table:
| Time period | Billed Amount (dollars) | Proportion of yearly mean | Deseasonalised data |
|---|---|---|---|
| \text{Feb 2018} | 139 | 0.7831 | |
| \text{Apr 2018} | 134 | 0.7549 | |
| \text{Jun 2018} | 269 | 1.5155 | |
| \text{Aug 2018} | 170 | 0.9577 | |
| \text{Oct 2018} | 125 | 0.7042 | |
| \text{Dec 2018} | 228 | 1.2845 | |
| \text{Feb 2019} | 70 | 0.6017 | |
| \text{Apr 2019} | 76 | 0.6533 | |
| \text{Jun 2019} | 179 | 1.5387 | |
| \text{Aug 2019} | 126 | 1.0831 | |
| \text{Oct 2019} | 54 | 0.4642 | |
| \text{Dec 2019} | 193 | 1.6590 |
Calculate the seasonal component for each billing period, to four decimal places:
Feb
Apr
Jun
Aug
Oct
Dec
Deseasonalise the data in the table. Round your answers to two decimal places.
Use your CAS calculator to calculate the equation of the least squares regression line for the deseasonalised data. Let t = 1 for Feb 2018. Give the equation of the line in the form y = a t + b and round any figures to two decimal places.
Predict the electricity bill amount for Feb 2021.
Comment on the reliability of your prediction.