MatStat#
Statistical testing#
Statistical testing is a formal procedure used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population.
In practice, it involves:
Formulating hypotheses:
The null hypothesis (\(H_0\)): a default assumption (e.g., “no effect”, “no difference”).
The alternative hypothesis (\(H_1\) or \(H_a\)): the assertion you want to test (e.g., “there is an effect”).
Calculating a test statistic from the data that summarizes the difference between what is observed and what is expected under the null hypothesis.
Determining a p-value, which tells you the probability of obtaining a result at least as extreme as the observed one, assuming the null hypothesis is true.
Making a decision:
If the p-value is below a predefined threshold (commonly 0.05), reject the null hypothesis.
Otherwise, do not reject it.
Statistical testing is widely used in scientific research, quality control, medicine, and many other fields.
Check more on statistical testing on the corresponding page.
Central limit theorem#
Here’s the improved version of your text:
The formulation of the central limit theorem is:
Let’s take \(N\) independent values \(x_i\) drawn from the same distribution with expected value \(\mu\) and standard deviation \(\sigma\).
We denote their sum as:
\(S_N = \sum_{i=1}^N x_i\)
The central limit theorem states that \(S_N \sim \mathcal{N}(\mu N, \sigma^2 N)\) — that is, \(S_N\) follows a normal distribution with mean \(\mu N\) and variance \(\sigma^2 N\).
Check more details in the corresponding page.