Curriculum links
The use of normal distribution to calculate and interpret expected values from practical situations. How data can be used to inform businesses of their success.
This series of linked activities covers the following:
• Unit Standard US5258
Use expected values to solve problems
• Statistics and Modelling 3.3 AS90643
Solve straightforward problems involving probability
• Statistics and Modelling 3.6 AS90646
Use probability distribution models to solve straightforward problems.
Mathematical processes:
• using and justifying mathematical models
• interpret information and results in context
• use words and symbols to describe and generalise patterns
• organising and interpreting data, using diagrams, graphs and models
• record information in ways that are helpful for drawing conclusions and making generalisations.
Information
The activities in StatZing! Secondary October 2006 use time-series analysis to make forecasts from historical data. The web activities make use of other information gathered at a accommodation establishment to analyse potential earnings. They involve a practical application of expectation algebra and normal distribution. The results can be used to analyse the occupancy rates and income generation of a fictitious accommodation establishment.
Together, the web and newsletter activities show how a commercial accommodation business could use their historical data to analyse the service that they provide, and to be informed about possible trends and potential profits. .
Answers
Measuring Commercial Accommodation
Part 1
1.
Events, e.g. the Ironman, Ellerslie Flower Show, Warbirds over Wanaka; ski season; families on summer holidays; New Zealand holidays, e.g. Waitangi Day, Labour Day, regional anniversary days; overseas visitors on holiday in our summer months; New Zealand’s growing domestic population.
2. (a) y = 1,900,000 approximately
(b) Select two coordinate points, e.g. (6, 1.9 million) and (48.2.4 million)
y = 11904.7619x + 1828500 [y = 11900x + 1829000 approximately]
(c) Select two coordinate points, e.g. (51, 2.49 million) and (60.2.5 million)
y = 1111.111x + 2433300 [y = 1110x + 2433300 approximately]
3. Trend – overall, the trend is for the occupancy rate of each type of accommodation to increase, though some are very small increases. The largest increase is for backpackers and hostels, which have been increasing steadily each year and are now close to hotels and motels.
In 2003, all types of accommodation, except hotels, showed a jump in occupancy. The drop in hotels may have been because there was a higher rate the year before.
4. Caravan parks and camping grounds have the lowest rate. This is probably because they are more seasonal and many are closed in winter.
Hosted accommodation is also lower. This may also be because some are not open at certain times of the year.
Backpackers are showing the largest increase. They are obviously increasing in popularity, possibly because of the low cost compared with other types of accommodation.
Occupancy rates
Part 2
For example, seasonal use of caravan parks and camping grounds, younger domestic overseas tourists and travellers using hosted and backpacker/hostel accommodation.
Season |
Occupancy rate |
Moving mean of 4 |
Centred moving mean |
Seasonal adjust- ment |
1998 Autumn |
0.598 |
|
|
|
Winter |
0.654 |
|
|
|
Spring |
0.845 |
0.73 |
0.73325 |
0.11175 |
1999 Summer |
0.823 |
0.7365 |
0.738125 |
0.084875 |
Autumn |
0.624 |
0.73975 |
0.741875 |
-0.11788 |
Winter |
0.667 |
0.744 |
0.745875 |
-0.07888 |
Spring |
0.862 |
0.74775 |
0.74525 |
0.11675 |
2000 Summer |
0.838 |
0.74275 |
0.744375 |
0.093625 |
Autumn |
0.604 |
0.746 |
0.745875 |
-0.14188 |
Winter |
0.68 |
0.74575 |
0.750625 |
-0.07062 |
Spring |
0.861 |
0.7555 |
0.757875 |
0.103125 |
2001 Summer |
0.877 |
0.76025 |
0.75925 |
0.11775 |
Autumn |
0.623 |
0.75825 |
0.75925 |
-0.13625 |
Winter |
0.672 |
0.76025 |
0.760875 |
-0.08887 |
Spring |
0.869 |
0.7615 |
0.760125 |
0.108875 |
2002 Summer |
0.882 |
0.75875 |
0.762125 |
0.119875 |
Autumn |
0.612 |
0.7655 |
0.767375 |
-0.15538 |
Winter |
0.699 |
0.76925 |
0.76925 |
-0.07025 |
Spring |
0.884 |
|
|
|
2003 Summer |
0.891 |
|
|
|
1. An upward/increasing long-term trend. Without adjusting for the seasons, the occupancy rate based on the trend line for the moving means is showing a 0.0022 (0.22%) growth rate per season.
2. Use the individual summer seasonal effects over this time period to estimate the seasonal effect for summer.
Summer |
0.084875 |
Autumn |
-0.11788 |
Winter |
-0.07888 |
Spring |
0.11175 |
0.093625 |
-0.14188 |
-0.07062 |
0.11675 |
0.11775 |
-0.13625 |
-0.08887 |
0.103125 |
0.119875 |
-0.15538 |
-0.07025 |
0.108875 |
Seasonal adjustment |
0.10403125 |
-0.13784 |
-0.07716 |
0.110125 |
3. 0.882 – 0.10403125 = 0.77796875 (≈ 0.778)
4. Autumn 2003 0.76925 + 3x 0.0022 – 0.13784 = 0.63801 (≈ 0.638)
Winter 2003 0.76925 + 4x 0.0022 – 0.07716 = 0.70089 (≈ 0.701)
Spring 2003 0.76925 + 5x 0.0022 + 0.110125 = 0.890375 (≈ 0.890)
Summer 2004 0.76925 + 6x 0.0022 + 0.10403125 = 0.88648125 (≈0.886)
5. You cannot predict too far into the future. This would be a prediction that is two years from the historic data, and the model may not continue as viewed.
6. y = 0.0022x+0.7294