13.2 Hypothesis testing steps

  1. Define a hypothesis
  2. Make a test statistic
  3. compute a p-value (to quantify the prob of having a value which is equal or more extreme than the t-test result)
  4. decide if to reject \(H_0\)

Step 1 is what we decide based on our investigation.

Step 2 is to construct a t-statistic, it summarize the relation with \(H_0\).

if: \[H_{0}:\mu_{t}=\mu_{c}\] we have a two sample test as we are searching values on the left and on the right of the t-test results \[T=\frac{\mu_{t}-\mu_{c}}{s\sqrt{\frac{1}{n_t}+\frac{1}{n_c}}}\] \[s=\sqrt{\frac{(n_{t}-1)s_{t}^2+(n_{t}-1)s_{c}^2}{n_{t}+n_{c}-2}}\] A large absolute value of the T-statistic is against the \(H_0\).

Step 3 is to compute a p-value, the probability of observing a value which is equal or more extreme than the observed value.

P-value is observing a T-stat which is equal or more extreme than the observed statistic

The p-value let’s us interpret the scale of out t-statistic absolute result.

The t-stat value is arbitrarily “LARGE”, the p-value rescale it to (0 to 1), in terms of probability to find an equal or more extreme value.

Step 4 is to identify if to reject \(H_0\) or fail to reject \(H_0\). The smaller the p-value is the stronger is the evidence AGAINST the NULL hypothesis.

  • Type I error reject \(H_0\) when \(H_0\) is TRUE
  • Type I error Rate is the prob of type I error
  • Type II error no reject \(H_0\) when \(H_0\) is FALSE
  • POWER of hypothesis is the prob of not making type II error

There is a trade-off between type I & type II error