Category: Statistics and Probability
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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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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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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…
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Transformations of Probability Distributions
I will share an important formula which is skipped over in almost all calculus classes. It is the transformation of variables formula for a probability density function (pdf). I will derive the formula, and use it to transform a few probability distributions. Let’s call this distribution X, not to be confused with the variable x…