Statistical Inference for the General Education student

From the opening and closing paragraphs:

Many mathematics departments around the country offer an introductory statistics course for the general education student. Typically these students come to the mathematics classroom with minimal skills in arithmetic and algebra. In addition it is not unusual for these students to have very poor attitudes toward mathematics.

With this target population in mind one can design courses of study, called statistics, that will differ radically depending on what priorities are held. Many people choose to teach arithmetic through statistics and thereby build most of the course around descriptive statistics with some combinatorics. Others build most of the course around combinatorics and probabilities with some descriptive statistics. Few courses offered at this level spend much time or effort on statistical inference.

We believe that for the general education student the ideas of statistical inference and the resulting decision rules are of prime importance. This belief is based on the assumption that general education courses are included in the curriculum in order to help students to gain an understanding of their own essence, of their relationship to others, of the world around them, and of how man goes about knowing.

If you inspect most of the texts on the market today, you will find that they generally require that a student spend approximately a semester of study of descriptive statistics and probability theory before attempting statistical inference. This makes it very difficult to get to the general education portion of the subject in the time allotted most general education courses. If you agree with the analysis of the problem to this point the logical question is ‘Is there a way to teach statistical inference without the traditional work in descriptive statistics and probability?’. The remainder of this article describes an approach that allows one to answer this question with a yes…

It should be pointed out that there are some unusual difficulties in this approach to statistics [since] one trades traditional weakness in arithmetic and algebra for deficiencies in writing since the write-ups of the simulations demand clear and logical exposition on the part of the student. However, if you feel that the importance of ‘statistics for the general education student’ lies in the areas of inference and decision rules, then you should try this approach. You will like it.

This article won the 1978 George Polya award for expository excellence. Several techniques described this article probably would be modified with modern computer simulation today, but are still worthy of reading.

Click to access 00494925.di020678.02p03892.pdf

Introduction to the Catalan numbers

From the YouTube description:

Alissa S. Crans, Associate Professor of Mathematics at Loyola Marymount University, introduces viewers to the Catalan numbers, which take on a variety of different guises as they provide the solution to numerous problems throughout mathematics.

More on the Catalan numbers can be found at MathWorld and at Wikipedia and at http://www-math.mit.edu/~rstan/ec/. This video is accessible to the general public, including gifted elementary school students.

From the European Girls Mathematical Olympiad

Source: https://www.facebook.com/maanews/photos/a.170415820418.120984.153302905418/10152497151470419/?type=1&theater

Pythagorean quadruplets

Source: https://www.facebook.com/photo.php?fbid=494942050639298&set=a.416585131808324.1073741827.416199381846899&type=1&theater

Proofs that the square root of 2 is irrational

Here’s a resource with over 25 different proofs for demonstrating that $\sqrt{2}$ is irrational: http://www.cut-the-knot.org/proofs/sq_root.shtml

Mathematician maze

Source: https://www.facebook.com/maanews/photos/a.170415820418.120984.153302905418/10152497214445419/?type=1&theater

Approximations

Source: http://www.xkcd.com/1047/

An infinite number of mathematicians walk into a bar

An infinite number of mathematicians walk into a bar. The first one orders a beer, the second one orders a half a beer, the third orders a quarter of a beer, and this trend continues on for some time.

After a while, the bartender gets fed up and hands them 2 beers, shakes his head and says, “You mathematicians just don’t know your limits.”

Functions that commute

At the bottom of this post is a one-liner that I use in my classes the first time I present a theorem where two functions are permitted to commute. At many layers of the mathematics curriculum, students learn about that various functions can essentially commute with each other. In other words, the order in which the operations is performed doesn’t affect the final answer. Here’s a partial list off the top of my head:

Arithmetic/Algebra: $a \cdot (b + c) = a \cdot b + a \cdot c$ . This of course is commonly called the distributive property (and not the commutative property), but the essential idea is that the same answer is obtained whether the multiplications are performed first or if the addition is performed first.
Algebra: If $a,b > 0$ , then $\sqrt{ab} = \sqrt{a} \sqrt{b}$ .
Algebra: If $a,b > 0$ and $x$ is any real number, then $(ab)^x = a^x b^x$ .
Precalculus: $\displaystyle \sum_{i=1}^n (a_i+b_i) = \displaystyle \sum_{i=1}^n a_i + \sum_{i=1}^n b_i$ .
Precalculus: $\displaystyle \sum_{i=1}^n c a_i = c \displaystyle \sum_{i=1}^n a_i$ .
Calculus: If $f$ is continuous at an interior point $c$ , then $\displaystyle \lim_{x \to c} f(x) = f(c)$ .
Calculus: If $f$ and $g$ are differentiable, then $(f+g)' = f' + g'$ .
Calculus: If $f$ is differentiable and $c$ is a constant, then $(cf)' = cf'$ .
Calculus: If $f$ and $g$ are integrable, then $\int (f+g) = \int f + \int g$ .
Calculus: If $f$ is integrable and $c$ is a constant, then $\int cf = c \int f$ .
Calculus: If $f: \mathbb{R}^2 \to \mathbb{R}$ is integrable, $\iint f(x,y) dx dy = \iint f(x,y) dy dx$ .
Calculus: For most differentiable function $f: \mathbb{R}^2 \to \mathbb{R}$ that arise in practice, $\displaystyle \frac{\partial^2 f}{\partial x \partial y} = \displaystyle \frac{\partial^2 f}{\partial y \partial x}$ .
Probability: If $X$ and $Y$ are random variables, then $E(X+Y) = E(X) + E(Y)$ .
Probability: If $X$ is a random variable and $c$ is a constant, then $E(cX) = c E(X)$ .
Probability: If $X$ and $Y$ are independent random variables, then $E(XY) = E(X) E(Y)$ .
Probability: If $X$ and $Y$ are independent random variables, then $\hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y)$ .
Set theory: If $A$ , $B$ , and $C$ are sets, then $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ .
Set theory: If $A$ , $B$ , and $C$ are sets, then $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ .

However, there are plenty of instances when two functions do not commute. Most of these, of course, are common mistakes that students make when they first encounter these concepts. Here’s a partial list off the top of my head. (For all of these, the inequality sign means that the two sides do not have to be equal… though there may be special cases when equality happens to happen.)

Algebra: $(a+b)^x \ne a^x + b^x$ if $x \ne 1$ . Important special cases are $x = 2$ , $x = 1/2$ , and $x = -1$ .
Algebra/Precalculus: $\log_b(x+y) = \log_b x + \log_b y$ . I call this the third classic blunder.
Precalculus: $(f \circ g)(x) \ne (g \circ f)(x)$ .
Precalculus: $\sin(x+y) \ne \sin x + \sin y$ , $\cos(x+y) \ne \cos x + \cos y$ , etc.
Precalculus: $\displaystyle \sum_{i=1}^n (a_i b_i) \ne \displaystyle \left(\sum_{i=1}^n a_i \right) \left( \sum_{i=1}^n b_i \right)$ .
Calculus: $(fg)' \ne f' \cdot g'$ .
Calculus $\left( \displaystyle \frac{f}{g} \right)' \ne \displaystyle \frac{f'}{g'}$
Calculus: $\int fg \ne \left( \int f \right) \left( \int g \right)$ .
Probability: If $X$ and $Y$ are dependent random variables, then $E(XY) \ne E(X) E(Y)$ .
Probability: If $X$ and $Y$ are dependent random variables, then $\hbox{Var}(X+Y) \ne \hbox{Var}(X) + \hbox{Var}(Y)$ .

All this to say, it’s a big deal when two functions commute, because this doesn’t happen all the time.

I wish I could remember the speaker’s name, but I heard the following one-liner at a state mathematics conference many years ago, and I’ve used it to great effect in my classes ever since. Whenever I present a property where two functions commute, I’ll say, “In other words, the order of operations does not matter. This is a big deal, because, in real life, the order of operations usually is important. For example, this morning, you probably got dressed and then went outside. The order was important.”

Calculators and complex numbers (Part 24)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

To begin, we recall that the trigonometric form of a complex number $z = a+bi$ is

$z = r(\cos \theta + i \sin \theta) = r e^{i \theta}$

where $r = |z| = \sqrt{a^2 + b^2}$ and $\tan \theta = b/a$ , with $\theta$ in the appropriate quadrant. As noted before, this is analogous to converting from rectangular coordinates to polar coordinates.

Theorem. If $z = x + i y$ , where $x$ and $y$ are real numbers, then

$e^z = e^x (\cos y + i \sin y)$

Definition. Let $z = r e^{i \theta}$ be a complex number so that $-\pi < \theta \le \theta$ . Then we define

$\log z = \ln r + i \theta$ .

Definition. Let $z$ and $w$ be complex numbers so that $z \ne 0$ . Then we define

$z^w = e^{w \log z}$

Technical point: for the latter two definitions, these are the principal values of the functions. In complex analysis, these are usually considered multiply-defined functions. But I’m not going to worry about this technicality here and will only consider the principal values.

This is the last post in this series, where I state some generalizations of the Laws of Exponents for complex numbers.

In yesterday’s post, we saw that $z^{w_1} z^{w_2} = z^{w_1 + w_2}$ as long as $z \ne 0$ . This prevents something like $0^4 \cdot 0^{-3} = 0^1$ , since $0^{-3}$ is undefined.

Theorem. Let $z \in \mathbb{C} \setminus \{ 0 \}$ , $w \in \mathbb{C}$ , and $n \in \mathbb{Z}$ . Then $(z^w)^n = z^{wn}$ .

As we saw in a previous post, the conclusion could be incorrect outside of the above hypothesis, as $\displaystyle \left[ (-1)^3 \right]^{1/2} \ne (-1)^{3/2}$ .

Theorem. Let $u \in \mathbb{R}$ and $z \in \mathbb{C}$ . Then $(e^u)^z = e^{uz}$ .

Theorem. Let $x, y > 0$ be real numbers and $z \in \mathbb{C}$ . Then $x^z y^z = (xy)^z$ .

Again, the conclusion of the above theorem could be incorrect outside of these hypothesis, as $(-2)^{1/2} (-3)^{1/2} \ne \left[ (-2) \cdot (-3) \right]^{1/2}$ .

green line

For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.