## A quick argument for why we don’t accept the null hypothesis

Introduction

When doing hypothesis testing, an often-repeated rule is ‘never accept the null hypothesis’. The reason for this is that we aren’t making probability statements about true underlying quantities, rather we are making statements about the observed data, given a hypothesis.

We reject the null hypothesis if the observed data is unlikely to be observed given the null hypothesis. In a sense we are trying to disprove the null hypothesis and the strongest thing we can say about it is that we fail to reject the null hypothesis.

That is because observing data that is not unlikely given that a hypothesis is true does not make that hypothesis true. That is a bit of a mouthful, but basically what we are saying is that if we make some claim about the world and then we see some data that does not disprove this claim, we cannot conclude that the claim is true.…

## p-values: an introduction (Part 1)

The starting point

This is the first of (at least) 3 posts on p-values. p-values are everywhere in statistics- especially in fields that require experimental design.

They are also pretty tricky to get your head around at first. This is because of the nature of classical (frequentist) statistics. So to motivate this I am going to talk about a non-statistical situation that will hopefully give some intuition about how to think when interpreting p-values and doing hypothesis testing.

My New Car

I want to buy a car. So I go down to the second hand car dealership to get one. I walk around a bit until I find one that I like.

I think to myself: ‘this is a good car’.

Now because I am at a second-hand car dealership I find it appropriate to gather some data. So I chat to the lady there (looks like a bit of a scammer, but I am here for a deal) about the car.…

## R-squared values for linear regression

Linear regression is a common and useful statistical tool. You will have almost certainly come across it if your studies have presented you with any sort of statistical problems.

The pros of regression are that it is relatively easy to implement and that the relationship between inputs and outputs is linear (it’s in the name, but this simplifies the interpretation of the relationship significantly). On the downside, it relies fairly heavily on frequentist interpretation of probability (which is a little counterintuitive) and it’s very easy to draw erroneous conclusions from different models.

This post will deal with a measure of how good a model is: $R^2$. First, I will go through what this value means and what it measures. Then, I will discuss an example of how reliance on  $R^2$  is a dangerous game when it comes to linear models.

What you should know

Firstly, let’s establish a bit of context.…

## Cantor–Schröder–Bernstein Theorem

Knowledge this posts assumes: What is a set, set cardinality, a function, an image of a function and an injective (one-to-one) function.

David Hilbert imagines a hotel with an infinite number of rooms. In this hotel, each room can only be occupied by one guest, and each room is indeed occupied by exactly one guest. What happens if more guests show up? Can they be accommodated for?

PAUSE: WHAT DO YOU THINK AND WHY?

Suppose we propose they cannot be accommodated for, since all the rooms are occupied. Hilbert then claims that he can define the functions $f:A \mapsto B,$ and $g:B \mapsto C,$ where $A$ is a set containing all current guests, and $f$ simply maps each guest to a room in the set $B$, and $g$ maps each room in $B$ to a new one in $C$. Notice that these functions must be injective, since if a room contains two different guests, those two different guests must be the same guest; recall $f(a) = f(b) \rightarrow a = b$.…

## 1.6 Partitions

Recall the  relation $\equiv \text{ mod} (4)$ on the set $\mathbb{ N}.$

One of the equivalence classes is $[0] = \{ ..., -8, -4, 0, 4, 8, ...\}$ which is equivalent to writing $[0] = [4] = [-4] = [8] = [-8] ...$

We could do this because the equivalence class collects all the natural numbers that are related to zero under the relation $\equiv \text{ mod} (4)$

The following theorem generalises this idea for any relation $\equiv \text{ mod} (n)$ on the set $\mathbb{ N}:$ for the integer $n.$

Let $R$ be an equivalence relation on set $A.$ If $a, b \in A,$ then $[a] = [b] \iff aRb.$

Essentially, equivalence classes  $[a] = [b]$ are equal if the elements  $a, b \in A,$ are related under the relation $R.$ And simultaneously, knowing that elements $a, b \in A,$ are related under $R$ means their equivalence classes  $[a] = [b]$ are equal.

An equivalence class  $\equiv \text{ mod} (n)$ divides set a $A$ into $n$ equivalence classes. We call this situation a partition of set $A.$

A partition of a set $A$ is defined as a set of non-empty subsets of $A,$ such that both these conditions are simultaneously satisfied:

(i) the union of all these subsets equals $A.$

(ii) the intersection of any two different subsets is

Let’s return to our example: $\equiv \text{ mod} (4)$ on the set $\mathbb{ N}.$ We could represent this set as:

• NOTE: Each equivalence class above represents an infinite set and despite the drawing suggesting $[0]$ is larger than $[3]$ for instance, this is not true.

## Review: Calculus Reordered

Book title: Calculus Reordered: A History of the Big Ideas
Author : David M. Bressoud

Princeton University Press
Link to the book: Calculus Reordered: A History of the Big Ideas

Discussions on the history of different fields are usually dry, wordy and generally, when you are studying the field, hard to read. This is because they are usually geared towards the general audience, and in doing so most authors tend to strip away the very exciting technical details. I expected the same treatment from the author, but I was pleasantly surprised.

The book contains $5$ chapters, which are the following:

1) Accumulations
2) Ratios of Change
3) Sequences of Partial Sums
4) The Algebra of Inequalities
5) Analysis

Each of these chapters has a central theme that is being covered, but they are not at all disjoint. For instance, the last three contain the history of concepts that would normally be found in a first course for Real Analysis, while the first two are essentially the more applied spectrum to serve as some form of motivation for going through all this trouble, although they can certainly stand on their own.…