Sample Solutions
Starting Point 1: Who’s Who in the Games.
Part A: A look at some Commonwealth Countries.
1. Countries were randomly selected by numbering the thirty countries that have participated in 10 or more games from 1 to 30. Random numbers from 1- 30 were generated using the RandInt(1,30) command on the calculator. Countries were selected, repeats and Australia ignored, until 19 were obtained.
The selected countries, including Australia were:
Country / Population / GDPAustralia / 20,090,437 (July 2005 est.) / $32,000 (2005 est.)
Bahamas / 301,790* / $18,800 (2005 est.)
Bermuda / 65,365 (July 2005 est.) / $36,000 (2003 est.)
Canada / 32,805,041 (July 2005 est.) / $32,800 (2005 est.)
Fiji / 893,354 (July 2005 est.) / $6,000 (2005 est.)
Gibraltar / 27,884 (July 2005 est.) / $27,900 (2000 est.)
Guyana / 765,283* / $3,900 (2005 est.)
Hong Kong / 6,898,686 (July 2005 est.) / $36,800 (2005 est.)
India / 1,080,264,388 (July 2005) / $3,400 (2005 est.)
Isle of Man / 75,049 (July 2005 est.) / $28,500 (2003 est.)
Kenya / 33,829,590* / $1,200 (2005 est.)
Malaysia / 23,953,136 (July 2005 est.) / $10,400 (2005 est.)
New Zealand / 4,035,461 (July 2005 est.) / $24,100 (2005 est.)
Papua New Guinea / 5,545,268 (July 2005 est.) / $2,400 (2005 est.)
Scotland / 5,120,000 / $30,900 (2005 est.)
Singapore / 4,425,720 (July 2005 est.) / $29,700 (2005 est.)
Tanzania / 36,766,356* / $700 (2005 est.)
Uganda / 27,269,482* / $1,700 (2005 est.)
Wales / 2,921,000 / $30,900 (2005 est.)
Zimbabwe / 12,746,990* / $1,900 (2005 est.)
2. (a)
2 (b) Two Outliers: Tanzania, population 36 766 356; and India, population 1 080 264 388.
2(c) The distribution is positively skewed, but with the third quartile of data more spread than the fourth quarter.
3. (a)
3 (b) The data is negatively skewed. The second quarter of data has the greatest spread. No outliers.
4. (a) sp = 11928665
sG = 13726
4. (b) Re-number the remaining countries 1 - 18 and use RandInt(1,18) to select 5.
Sample is 4, 18, 14, 6, 5
Country / Population / GDPCanada / 32,805,041 (July 2005 est.) / $32,800 (2005 est.)
Fiji / 893,354 (July 2005 est.) / $6,000 (2005 est.)
Gibraltar / 27,884 (July 2005 est.) / $27,900 (2000 est.)
Papua New Guinea / 5,545,268 (July 2005 est.) / $2,400 (2005 est.)
Uganda / 27,269,482* / $1,700 (2005 est.)
4.(c) (i) Canada zG = 0.95
Fiji zP = -0.77 zG = -1.00
Gibraltar zP = -0.84 zG = 0.59
PNG zP = -0.38 zG = -1.27
Uganda zP = 1.44 zG = -1.32
4(d) Three countries (Fiji, Gibraltar and PNG) fall within the middle 68% of the sample for Population. Canada and Uganda are in the top 16% of countries in population terms.
Canada and Gibraltar are in the middle 68% of countries in the sample in terms of GDP. Fiji, PNG and Uganda are in the lower 32% of countries in GDP terms.
Part B: The bigger the Better.
1.
2.
M = Number of medals won. A = Number of participating athletes.
M = 0.39A - 7.66
3. The scatterplot (or more appropriately a residual plot for the linear model) suggests that an x2 transformation might be suitable.
For the x2 transformation,
Note: For a 1/y transformation, the calculator, rightly, refuses to divide by zero!!
4. The x2 transformation is a better fit for the data, with r2 = 0.936 compared with 0.913 for the linear x model. Hence the better equation to describe the data is:
M = 0.39A2 - 7.66
5.
The residuals show an increasing loss of accuracy for the model. It appears to be most accurate for smaller values of A2.
And......
Part C: Improving or Not?
1.
2. England and Australia demonstrate a reasonably consistent upward trend. Canada decreased from 1930 to around 1960 and has since shown an upward trend. Overall Australia has the greatest upward trend, particularly since 1982. Spikes, such as in 1932 for Australia, often correspond with years in which the country was the host nation.
3.
Starting Point 2: Winners are Grinners.
Part A: How Good is Australia?
1. (a) The countries are numbered 1 - 72. Twenty four random numbers between 1 and 72 are made using the calculator function, RandInt(1, 72). Repeated numbers, and 4 (Australia) are ignored and selections continued until 24 countries have been selected.
COUNTRY / AV NO OF MEDALS/GAMESAden / 0
Anguilla / 0
Australia / 99
Bangladesh / 1
Bermuda / 0
Cayman Islands / 0
Cook Islands / 0
Dominica / 1
Guernsey / 1
Hong Kong / 2
Isle of Man / 1
Kiribati / 0
Lesotho / 0
Montserrat / 0
New Zealand / 29
Nigeria / 13
Niue / 0
Pakistan / 9
Saudi Arabia / 0
Seychelles / 1
South Africa / 27
St. Lucia / 0
Swaziland / 0
Tonga / 0
Vanuatu / 0
STEM / LEAF
0 / 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 2 9
1 / 3
2 / 7 9
3
4
5
6
7
8
9 / 9
1 (b)
1. (c) Lowest value = 0
Q1 = 0
Median = 0
Q3 = 1.5
Highest Value = 99
The data is strongly positively skewed. All values above 2 are outliers.
2. (a)
2 (b) This data is also strongly positively skewed. The Games appear to be dominated by a small number of very powerful countries. Frequent participation does not appear to improve the relative performance of smaller countries.
Part B: Gold Medal Winners
1 (a)
GDP ($US MillionsAUSTRALIA / 631,256 / GHANA / 8,620 / FIJI / 2,627
ENGLAND / 1,790,000 / NAURU / 65 / BARBADOS / 2,627
CANADA / 979,764 / CAMEROON / 14,733 / NAMIBIA / 5,456
NEW ZEALAND / 99,687 / UGANDA / 6,833 / ISLE OF MAN / 2,138
SOUTH AFRICA / 212,777 / TRINIDAD / 12,544 / MOZAMBIQUE / 5,548
INDIA / 691,876 / BAHAMAS / 5,260 / BANGLADESH / 56,844
SCOTLAND / 184,000* / SINGAPORE / 106,818 / GUERNSEY / 2,609
KENYA / 15,600 / ZIMBABWE / 17,750 / PNG / 3,909
WALES / 62,100* / CYPRUS / 15,418 / BERMUDA / 2,353
NIGERIA / 72,106 / TANZANIA / 10,851 / MAURITIUS / 6,056
JAMAICA / 8,030 / HONG KONG / 163,005 / JERSEY / 3,632
MALAYSIA / 117,776 / ZAMBIA / 5,389 / LESOTHO / 1,375
NORTHERN IRELAND / 62100* / GUYANA / 786 / ST VINCENT / 403
PAKISTAN / 96,115 / SRI LANKA / 20,055 / ST KITTS & NEVIS / 397
For the upper values,
For the middle values,
For the lower values,
1 (b) i The upper set of values has the greatest spread and is strongly positively skewed, with one outlier. The middle set of data is more symmetrical, but with two significant outliers. The lower set of data is unevenly distributed with one significant outlier.
ii In general terms, yes, but exceptions exist.
2.
The data spreads very widely as x (GDP) increases. The correlation co-efficient is 0.84. A linear model can be tried.
2 (b)
Three median Regression Line: y = 0.0004x + 0.87
(c) Least Squares Regression Line: y = 0.00035x +8.3
It is doubtful that either model is particularly useful, given that the gradients of both are almost zero. The three median line would have an advantage in not being so effected by outliers.
Part C: Are We Getting Better?
1.
2.
3. Both the total number of medals available at each games, and the number won by Australia are trending upwards. The percentage of medals won by Australia at the Games shows no trend.
Starting Point 3: Breaking Records
Part A: Australia to the Fore.
1 (a)
00 / 6 / 8
1 / 2 / 3 / 4
1 / 7 / 7 / 9
2 / 1 / 1 / 2
2 / 5 / 9
3 / 0 / 3 / 3 / 3 / 3 / 3
3 / 6 / 8
4 / 0 / 1
4 / 7
5 / 0
5
5│0 = 50%
1 (b) The results are unevenly distributed, tending towards bi-modal.
1 (c) The results are not greatly indicative of performance in each Sport. The data is markedly affected by some very small values (eg Water Polo, Cricket, and the fact that a maximum percentage of 33% is achievable for team events, 100% for individual events.
2. (a)
2 (b) England appears to be strong in Archery. Canada appears to be strong in Triathlon, but only two events have been held. Australia outperforms the other two countries in all sports except Archery, Rowing (equal best) and Triathlon.
Part B: Success Breeds Success.
1. (a) (b)
1 (c) Sports above the line suggest more records than predicted; Weightlifting being the standout.
2 (a)
2 (b) Possible transformations are log(x) and.
Log(x) transformation: y2 transformation:
Part C: Faster, Higher Longer
1.
2. Both men’s and women’s results show an upward trend. The women’s results have a greater overall rate of improvement.
3.
4.
That is, L = 0.016(Y) + 7.24 where L = distance (metres), Y = Years since 1930.
The predicted winning distance in Melbourne:
L = 0.016(Y) + 7.24
= 0.016(76) + 7.24
= 8.46m