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Hypothesis Testing and p-values
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Formulating null and alternative hypotheses H0 and H1, then using the p-value from the data to judge evidence against H0 at a chosen significance level.
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Null and alternative hypotheses (H₀ & H₁)
SL AI 4.11
When we want to make a claim using statistics, need sufficient evidence. Flipping a coin and getting 3 heads in a row is not strong evidence that it is biased, but 100 in a row is.
Whatever data we have, we start by assuming that they are produced by random chance alone. We call this the null hypothesis, which we write H0. In the coin flip example, the null hypothesis is H0: the coin is fair.
An alternative hypothesis, denoted H1, is the idea that something "fishy" is going on. In the coin flip example, this could be H1: the coin is biased towards heads.
It's important (both in exams and real life) to assume the null hypothesis is true unless you have good evidence.
Writing down the null and alternative hypotheses can be hard, but you can think of H0 as a neutral assumption, and H1 as something we need evidence to prove.
More Examples
Does listening to music while studying hurt test performance?
Null hypothesis H0: we assume it makes no difference: the average scores of students with or without music are similar
Alternative hypothesis H1: students who listen to music do worse: they have a lower average test score.
Does drinking an energy drink improve reaction time?
Null hypothesis H0: we assume it makes no difference: the average reaction times with and without energy drinks are similar.
Alternative hypothesis H1: drinking an energy drink lowers the mean reaction time.
Significance Levels & p-values
SL AI 4.11
Once we have our null and alternative hypotheses, we use our data as evidence against the null hypothesis.
Let's take the coin flip example, and start by assuming the null hypothesis: it is fair. That means each time I flip it, I have a 21 probability of getting heads. If the coin gives 10 heads in a row, the probability is
(21)10≈0.098%
This number is the probability of the data we observed assuming the null hypothesis. The smaller it gets, the less likely that the null hypothesis is true.
We call this the p-value. The smaller the p-value, the stronger the evidence for the alternative hypothesis. If the p value is less than the significance level α, we reject the null hypothesis, which is essentially concluding the alternative hypothesis is true.
p-value: The probability of getting results as surprising (or more) as the observation if the null hypothesis were true.
Significance level (α): The cutoff we choose in advance. If the p-value is below α, we reject the null hypothesis.
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