There is literally no area in which tests of hypotheses do not find application. Though the methodology was put on a firm footing in the 1930's, the very first known application of tests of hypotheses, using the logic that is still the basis for them to this day, is from 1710. That year, John Arbuthnott, a Scottish polymath, who, in addition to knowing/being friends with a variety of bigwigs such as Jonathan Swift, Isaac Newton and Samuel Johnson, was also the Royal Physician to Queen Anne, published a paper in the Philosophical Transactions titled Arbuthnott stated in his paper that for 82 years (1629-1710) in a row in London, there were more male babies born every year than female babies. He made this observation by showing a table (below) in the paper that shows the number of male and female christenings in London; this clearly is a leap of faith because the number of christenings would be substantially less than the actual number of births at that time, due to causes such as a much higher infant mortality rate, the stigma surrounding illegitimate babies, etc. Nevertheless, we will let this slide (and treat # of christenings as equal to # of births) and continue with Arbuthnott's argument, which is as follows, and the logic of which continues to be used to this day: Arbuthnott first assumed that there is no Divine Providence guiding the world and hence, since only Chance governed everything by this assumption, he concluded correctly that P(Baby is M) = P(Baby is F)=0.5 (this is like tossing a fair coin)

From here, he argued that the implication of the above probability was that P(# of Males born in a year > # of Females born that year) 0.5 to a very good approximation and so set this probability to be exactly 0.5 (The Binomial distribution method, which we will not cover in this class, is used to prove this approximation. However, we will not worry about this but will take the approximation for granted in what follows.). Next, he answered the following question: Given the assumption above and the subsequent calculations, what is the probability that there are more Males born than Females each year, for 82 years in a row? Note that this event corresponds to the joint event (# M born yr 1 > #F born yr 1) &(# M born yr 2 > #F born yr 2) &...&(# M born in yr 82 > #F born in yr 82)

a) Compute the above probability, showing the steps and assumptions you make to get your answer. Do you think that any assumptions you may have made are reasonable and, if yes, why? (Note: This prob computed in this context, we shall see later, is an example of what is called the p-value)

b) Note that the probability you computed in (a) was based on the foundational assumption that P(M)=P(F)=0.5. Given this probability of the event you computed in (a) with the FACT that more births were male than female in every single year for 82 consecutive years, what do you feel about the foundational assumption that P(M)=P(F)=0.5? (Note: This foundational assumption in this context, we will see later, is an example of what is called the Null Hypothesis)