Insight Things

A scientific blog revealing the hidden links which shape our world

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Power of Statistical Tests – The Principles

I guess that you already know a little bit about hypothesis testing. For instance, you might have carried out tests in which you tried to reject the hypothesis that your sample comes from a population with a hypthesized mean ยต. As you know that the sample mean follows a t-distribution (or the normal distribution in case of huge samples), you can define a rejection region based on a specific significance level ฮฑ. In a one-sided test, sample means which are less than a critical value CV might be considered to be rather unlikely. If the obtained sample’s mean falls into this region, the hypothesis gets rejected at this particular signficance level ฮฑ.

Distribution of sample mean and rejection region for one-sided test

Distribution of sample mean and rejection region for one-sided test

We know the chance of rejecting the hypothesis although it’s true (Type I error), because it is the chance of obtaining just one of those values from the rejection region (plotted red). But how likely are we to reject a hypothesis if it’s indeed false? In other words: How small is the type II error? more โ†’

Toilet Paper Physics – The Art of Tissue

One day when I spent some day in the smallest room of my company, I really got annoyed by the toilet paper getting ripped off after the first piece of paper. Maybe you have also experienced this, especially when a toilet paper dispenser like the one shown below is involved.

Toilet paper dispenser showing unpleasant behaviour

Toilet paper dispenser showing unpleasant behaviour

There is kind of an unwind impediment integrated in these dispensers which makes you pull hard on the loo paper. That’s why it becomes so evident that the position where the toilet paper breaks is influenced by how you pull it. Anyway, the following explanations account for usual toilet paper consumption as well ๐Ÿ˜‰ more โ†’

Integration by Substitution – Why it works!

Integration by substitution (often referred to as u-substitution) is quite hard to understand. Most people just follow their proven recipe when performing this kind of integration. In contrast to that doing, I want to illustrate why the steps of u-integration are necessary. Therefore we will develop the idea slowly by giving simple examples which illustrate what works and what doesn’t. more โ†’

Chi-squared distribution and normal variance

Those of you who already have tested for the variance of data from a normal distribution may have asked themselves how the link between normal variance and chi-squared distribution arises. Trust me: The story, which I will tell you, is an exciting one! more โ†’

Yoast SEO: Working around the meta-description issue

I really like the Yoast SEO plug-in for WordPress blogs. Unfortunately, the latest version causes trouble when it comes to inserting the meta description. In this post I will show you, how to set a meta description without rolling back to a previous version of Yoast SEO plug-in. more โ†’

High fuel consumption in winter

My dad told me that a car’s fuel economy is bad in winter, because the engine has to heat a lot more. Well, I read about cold engines, low tire pressure, higher rolling resistance and so on. They didn’t convince me. All together have a substantial impact,ย  I admit, but the most important factor of fuel consumption is not discussed sufficiently! more โ†’

Wind strength and fuel consumption

Imagine you drive in your car against head wind and later you return with tail wind. When asked, most people tend to say that the force of the wind on average equals the situation when there is no wind at all. This is surprising since the same people will tell you that the air resistance (the aerodynamic drag) increases as the squared velocity increases. I will show you why wind is substantial when discussing causes of high fuel consumption! more โ†’

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