统计代写|贝叶斯分析代写Bayesian Analysis代考|STATS 3023

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|贝叶斯分析代写Bayesian Analysis代考|Patterns and Randomness: From School League Tables to Siegfried and Roy

Take a look at Table 2.3. It shows the scores achieved (on an objective quality criteria) by the set of state schools in one council district in the UK. We have made the schools anonymous by using numbers rather than names. School 38 achieved a significantly higher score than the next best school, and its score (175) is over $52 \%$ higher than the lowest ranked school, number 41 (score 115). Tables like these are very important in the UK, since they are supposed to help provide informed “choice” for parents. Based on the impressive results of School 38 parents clamour to ensure that their child gets a place at this school. Not surprisingly, it is massively oversubscribed. Since these are the only available state schools in this district, imagine how you would feel if, instead of your child being awarded a place in School 38, he or she was being sent to school 41 . You would be pretty upset, wouldn’t you?
You should not be. We lied. The numbers do not represent schools at all. They are simply the numbers used in the UK National Lottery ( 1 to 49). And each “score” is the actual number of times that particular numbered ball had been drawn in the first 1,172 draws of the UK National Lottery. So the real question is: Do you believe that 38 is a “better” number than 41 ? Or, making the analogy with the school league table more accurate:

Do you believe the number 38 is more likely to be drawn next time than the number 41 ? (Since thé üsuall interpretatiòn of the schoool léaguee tablé is that if your child attends the school at the top he or she will get better grades than if he or she attends the school at the bottom.)

The fact is that the scores are genuinely random. Although the “expected” number of times any one ball should have been drawn is about 144 you can see that there is a wide variation above and below this number (even though that is still the average score).

What many people fail to realise is that this kind of variation is inevitable. It turns out that in any sequence of 1,172 lottery draws there is about a $50 \%$ chance that at least half the numbers will be chosen either less than 136 times or more than 152 times. That indeed is roughly what happened in the real sample. Moreover, the probability that at least one number will be chosen more than 171 times is about $45 \%$. You may find it easier to think of rolling a die 60 times. You would almost certainly not get each of the six numbers coming up 10 times. You might get 16 threes and only 6 fours. That does that not make the number three “better” than the number four. The more times you roll the die, the closer in relative terms will be the frequencies of each number (specifically, if you roll the die $n$ times the frequency of each number will get closer to $n$ divided by 6 as $n$ gets bigger); but in absolute terms the frequencies will not be exactly the same. There will inevitably be some numbers with a higher count than others. And one number will be at the “top” of the table while another will be “bottom.”

统计代写|贝叶斯分析代写Bayesian Analysis代考|Why You Should Be Very Wary of Correlations and Their Significance Values

The close “correlation” between per capita cheese consumption and number of people who died by becoming tangled in their bedsheets in the United States, as shown in Figure 2.12, is one of many spurious correlations published on the website http://tylervigen.com.

Correlations and significance values (also called p-values) are the standard techniques that statisticians use to determine whether there are genuine relationships between different variables. In the approach to probability that we espouse in this book these traditional techniques (along with their first-cousins regression analysis and confidence intervals, which we will look at in Chapter 12 ) are superseded by techniques that we feel are simpler and more intuitive. But the acceptance and entrenchment of these ideas are so widespread across all empirical disciplines that you need to be aware of what they are in order to appreciate the damage they can do to rational decision making and risk analysis.
Look at Table 2.6. This gives (a) the average temperature and (b) the number of automobile crashes resulting in fatalities in the United States in 2008 broken down by month (source: U.S. Department of Transport 2008). We can plot this data in a scatterplot graph as shown in Figure 2.13.

From a quick view of the chart there is a relationship between temperature and fatalities. There seem to be more fatalities as the temperature increases. Statisticians use a formula-called the correlation coefficient (see Box 2.3) – that measures the extent to which the two sets of numbers are related. You do not need to know what this formula is because any spreadsheet package like Excel will do the calculation for you. It so happens that the correlation coefficient in this case is approximately $0.869$. Ilsing standard tahles this turns nut to he “highly significant” (comfortably passing the criteria for a $p$-value of $0.01$ that is also explained in Box $2.3$ ). Statisticians would normally conclude from this data that the number of road fatalities and the minimum temperature on any given day are significantly related (although note that we have severe concerns about the limitations of $p$-values as explained in Box 2.3).

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