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How does AP Stats teach chi-squared tests in an imprecise way? How can the precision of the tests be improved?
This is my second post where I find issues with the methods taught in AP Stats. This time, it’s with the chi-squared test. When I took the class over a year ago, the final project was to conduct a statistical experiment and achieve a p-value that meets all of the requirements. In my experiment, I…
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Eating a balanced diet using linear algebra, Part 2
Last week, I reduced a matrix to solve for the amount of each food you need to eat to reach various nutrient goals. Unfortunately, 4/10 of the variables were negative, which makes no sense in the real world because you can’t eat negative food. Today, I will post a workaround I have for this issue.…
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Eating a balanced diet using linear algebra
An interesting application of linear algebra is using it to find out what you need to eat to reach your nutrient goals and how much. We will assemble an augmented matrix to model this. Each row of the matrix corresponds to one equation, e.g. x+2y+3z=22. The variables x, y, and z are three types of…
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Simplifying nested radicals
Nested radicals appear in many math problems, especially in trigonometry. They are expressions that have a root inside of another root, usually of the form √(x+√y). Example: This is quite messy to look at, but under certain circumstances it can be simplified into two non-nested radicals. This example can be reduced to: We can derive…
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How does AP Statistics teach confidence intervals for proportions incorrectly? What is a more refined method?
When I took AP Statistics during my sophomore year, the class taught how to construct a confidence interval for a proportion using a normal approximation to a binomial distribution. It was also called a “z-interval”. I don’t like it because it uses an approximation which limits the versatility of the method. The confidence interval is…
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How to graph a polynomial of degree n that goes through any n+1 points
A very long time ago, my math teacher told me that a quadratic could pass through any 3 points, a cubic any 4 points, etc. While this seemed intuitive to me, I could not think of a formula. About a year ago, I spent two hours deriving a general formula for the quadratic case, which…
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What happens when you take the derivative of the quadratic formula? And what does it mean?
In this post, I will discuss the meaning of the three partial derivatives of the quadratic formula, with respect to each variable (a, b, and c). This is a pretty old concept of mine (I first thought about this about 6 months ago). It was also inspired by a video by blackpenredpen. However, he only…
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Math problem originating from random image I saw online
Two days ago, someone posted the cover image in one of the Discord servers I’m in. While it is meant to be a joke about the ambiguity of this definition of a square, it made be genuinely curious about one thing: If the common side length is 1, and the segments of arc come from…
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Formula that produces normally distributed values from uniform random variables
A few months ago, I learned this formula: R1 and R2 are independent uniformly distributed random variables between 0 and 1, and N(0,1) is a normal distribution with a mean of 0 and a standard deviation of 1. I am going to prove that this formula actually gives normally distributed values using a few other…
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What is the meaning of an undefined mean?
The formula for the mean of a probability density function (pdf) f(x) defined on the interval (a,b) is the following: If this seems confusing, here is an interpretation of the formula. The formula multiplies the value (x) with the probability of finding that value in an infinitesimal sliver of the distribution (f(x)dx) and integrates it…