In 2014, 11 out of 12 months of the Netherlands were warmer than the average from 1981 to 2010:
Could this have happened by coincidence? Or does this mean that the climate in the Netherlands has become warmer?
To answer questions like this we can use the statistical technique of Hypothesis testing.
The first step in Hypothesis Testing is being precise in what the hypothesis is that you are testing. In this case, you want to consider the two possibilities:
We need to formulate this explicitly enough so we can test it in an experiment.
Formally, we do this by creating the null hypothesis $H_0$ and the alternative hypothesis $H_1$ (corresponding respectively to the 2 options above).
Let’s say a month in 2014 is warm if is warmer than the average for that month from 1981-2010. In this case we will take. Inn this case, we will take as null hypothesis $H_0$ the statement that the probability that a month is warm equals $\frac{1}{2}$. The null hypothesis should always corresponds to a default: nothing special is happening here.
The alternative hypothesis $H_1$, is the statement that the probability of a warm month is $>\frac{1}{2}$. If we can conclude that $H_1$ is true then we truly have shown something remarkable.
Technically you should formulate $H_0$ in such a way that you can calculate probabilities assuming $H_0$ is true. In this case, we need to add add the assumption that the probabilities that different months are warm are independent from each other. Then we can calculate the probability that our observation of $11/12$ months are warm would occur if $H_0$ were true.
After formulating our hypothesis, we perform the experiment. You have to do the experiment after formulating the hypothesis! Even random data always satisfies some peculiar pattern and we human beings are conditioned to look for such patterns. You first look at the data, you might identify a pattern that exists only by chance. If you test for this pattern which you already saw in the data and subsequently proof it is there, you have just proven that you are not blind.
You have not shown that the pattern is an intrinsic feature of the experiment you performed. In this case, we looked at the data first, so we are in trouble. We will just pretend we formulated this hypothesis before 2014. As there are quite a few scientists who claim that the earth is warming up, it would not have been so strange to pretend this.
After performing the experiment, you have to check whether the results are sufficient to reject $H_0$ and assume $H_1$ is true instead. Here we truly take $H_0$ to be the default position. If the results obtained could have happened with reasonable probability assuming $H_0$ is true, then we do not reject $H_0$. Only if the results are so outlandish that the odds of them happening if $H_0$ were true are vey small, then we reject $H_0$ and conclude that $H_1$ is likely valid.
In this case, evidence for $H_1$ consists of many warm months. Thus we would make a rule saying: if there are at least $x$ warm months in 2014, then we reject $H_0$, otherwise we do not reject $H_0$. Here $x$ is some number, which should have been determined before doing the experiment. Are 9 warm months already sufficient, or would you like to see at least 10 or even 11? We will later see how to determine $x$.