When will we see people of negative height? Group 1: Walter (Presenter) Jun Yue Celestine Crux of the issue: The distribution of the heights of human beings are believed to have a normal distribution. According to the empirical rule, this would mean that it is possible to observe human beings with negative heights. However, we have never observed anyone with negative height but we cannot be sure that people with negative height has never existed before. Hence, an experiment or research study must be done to find out if our heights indeed follow a normal distribution. We need sufficient subjects to collect a more complete or representative observations, and that will take at least 13000 years later.

Outline Normal Distribution to model height. Empirical Rule What is the probability of there being a fully grown adult with negative body height among people living today? What is the probability of there having been a fully grown adult with negative body height among people who have ever lived on earth? How many people are necessary to have ever lived on earth in order to observe at least 1 fully grown human with negative body height with a probability of at least 95%? By when can we expect to reach this necessary number of people ever lived on earth? (time frame?) Take Home Message Outline Basically i will talk on how normal distribution is used to model the height of human beings And explore the 4 questions raised in our reading namely…. Finally i will conclude with a take home message

Normal distribution to model human height So as we all know a normal distribution curve looks like this…. Like an upside down U. symmetric about the mean. Our reading starts off by telling us something most of us already know which is that…. Convention is to use normal distribution to model heights of human beings Our reading is questioning if the normal distribution the correct one for describing and analysing the heights of human beings There are people who agree that the normal distribution is the correct one for describing and this is because the observed data fits well and it is basically useful However, there are those who disagree and feel that the normal distribution is not the correct model because it means that there will exist people with negative height.

Empirical Rule 68.26% of the population lie within 1 SD of the mean 95.44% lie within 2 SDs of the mean 99.73% lie within 3 SDs of the mean 0.27% are more than 3 SDs from the mean. Since we are curious to find out about the extreme case of negative height we look at what proportion lies 18.5 SDs away from the mean? The answer to that is 1.06 x 10^-76. This is equivalent to say 1 person in 10^76 people will have zero or negative height So now we will look at what a normal distribution tells us about human height. Our reading introduces the empirical rule What proportion is 18.5SDs away from the mean? 1 in 10^76

As we can see here, most of the data would lie within 3 standard deviations from the mean of the data. So put in context, the height of most people would range from 1.490m to 2.066m. (if professor asks how we know Correlation variation = standard deviation / mean The reading takes the CV value to be 5.4% as the upper bound and mean height of men of 1.778 m as the upper bound So by manipulation → SD = 9.6cm = 0.096m The minimum of the range = -3x 0.096m + 1.778m = 1.490m (3sf) The maximum of the range = 3 x 0.096m + 1.778m = 2.066m (3sf)

What is the probability of there being a fully grown adult with negative body height living among us today? The world population at the time the article was written : about 7 billion (7 x 10^9) Fully grown adults: 5.17 billion (5.17 x 10^9) To get the probability we take the population of fully grown adults divide by 10^76 and it will give us something very small which is often rounded off to zero (5.33x 10^-67) Equivalent to saying 1 chance in 2.33 x 10^66 of seeing a person with negative body height

What is the probability of there having been a fully grown adult with negative body height throughout history? Number of people who have ever lived on Earth: 107.6 billion Upper estimate of the probability = 1.11 x 10^-65 We cannot reject our hypothesis We cannot be sure that cases of negative height in the early times of mankind would be recorded and come to our knowledge There may have been a lot of cases of negative height but these people went extinct due to evolution Equivalent to saying 1 chance in 10^-65 of seeing a person with negative body height We take the 107.6 billion as the upper boundary because we don’t know the number of adults in the histroy that has grown to adulthood and full height. Itmight even be that negative body height was nothing rare in the very early days of mankind but that positivebody height was an evolutionary advantage and therefore this characteristic became extinct.

What is the number of people needed for us to see at least 1 fully grown adult with negative body height with a probability of at least 95%? Answer: 2.9 × 10^76 (in words is 29 thousand billion billion billion billion billion billion billion billion people) Equivalent: A computer of the current generation, would need 8.75 × 10^52 years just to count to this number. Also much longer than the age of the earth (estimated to be around 4.6 billion years). The Big Bang is estimated to have occurred about 13.7 billion years ago. Writing the number is easy, but to comprehend it is not. Therefore relate/contextualise the number to the real world situation: computer counting, age of the earth, and the big bang

How much time do we need before we can have a large enough sample size to observe this phenomenon? Under certain assumptions we can estimate the time needed to have a large enough sample size. -Using growth rate of the global population, starting global population, life expectancy. -We have to account for life expectancy because every individual contributes in each year of his lifespan to the number of people alive. After factoring everything in and doing the calculations the answer is: 13842 YEARS FROM NOW

Take Home Message “All models are wrong, but some are useful - for part of their range at least.” - George Box So we should not totally dismiss all models because we can still use and learn things from them