ST440/540 – Mid-term exam 1
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ST440/540 – Mid-term exam 1
THIS IS AN EXAM - DO NOT DISCUSS THE PROBLEM WITH ANYONE (INCLUDING OTHER STUDENTS OR THE TA)! If you have questions, please email me.
The Olympics are for mathletes too. In this exam you will explore the home-country advantage in the Summer Olympics by the rate of medals per participant in the host year compared to their previous Olympics. The data1 are:
|
Host country Year |
Medals won Participants Previous Host Previous Host |
|
|
Finland 1952 Australia 1956 Italy 1960 Japan 1964 Mexico 1968 West Germany 1972 Canada 1976 Soviet Union 1980 United States 1984 South Korea 1988 Spain 1992 United States 1996 Australia 2000 Greece 2004 China 2008 Great Britain 2012 Brazil 2016 Japan 2021 |
35 36 29 9 40 11 195 174 33 22 101 58 16 100 65 19 51 |
294 280 328 275 423 385 489 522 401 422 647 617 426 599 530 462 621 |
Let Yi1 be the number of medals won by the host country during the Olympics i and Yi0 be the numer of medals won by the county in the previous Olympics. Similary, let Ni0 and Ni1 be the number of participates from the country in the corresponding Olympics. For example, Finland hosted the Olympics in 1952. They had N11 = 258 partipicants in 1952 and N10 = 129 participants in the 1948; they won Y11 = 22 medals in 1952 and Y10 = 24 medals in the 1948.
Analyze these data in the following sections:
1. Aggregate analysis: Combine data across all years for the host county in the host year, Y1 = P
Yi1 and N1 = P
Ni1. Conduct a Bayesian analysis of λ1, the ex- pected number of medals per participant in their home country. State a reasonable likelihood, an uninformative conjugate prior distribution and give the posterior distri- bution. Repeat this analysis using the data from the previous year, Y0 = P
Yi0 and
N0 = P
Ni0, to estimate λ0. Compare these two posterior distributions in a figure. What are the main assumptions in your analysis and do you think they are valid?
2. Hypothesis test: Conduct a Bayesian test of the hypothesis that there is a home- county advantage, i.e., λ1 > λ0. Clearly state your hypotheses and describe methods you are using for the test. Are your results sensitive to the prior?
3. Prediction: The next Olympics will be held in France in 2024. In 2021, France had
398 partipicants and won 33 medals (we do not know the number of participants in
2024 so you will have to predict this value). Predict the number medals France will win in the 2024 Olympics and quantify your uncertainty about this prediction. Clearly describe the methods you are using to make this prediction.
4. Country-specfic analysis: Conduct an analysis separately by country (combine the data across the two Olympics for AU, Japan and USA, so there are a total of 15 counties). Describe the likelihood, prior and posterior. Compare the posterior distri- bution of the ratio r = λ1/λ0 for each county. Is there evidence that the home-country advantage differs by country?
5. Conclusions: Summarize your main findings and enumerate at least two limitations to the your analysis, and suggest how these could be addressed in future work.
Your paper should be written as a professional document with full sentences and para- graphs, clearly labeled and numbered figures and tables, and few spelling/grammar errors. Organize your report with five sections, labelled as above. You should include enough detail that another student in class could reproduce your results. Summarize your analysis in a PDF document that is no more than four pages long (excluding code). Append your code to the end of this document and submit a single document on moodle.
2022-02-18