The p-value is the smallest significance level at which a test would have led to a critical region, given the observed statistic, ${\alpha }_0$

• testing the level of "extremicity", not just the values itself

• $x=[0,20]$, $\mu =10$ $\to$ p-value = $2\cdot P(X\le 4|H_0)$ (extremities in all directions)

• a measure of the strength of the evidence against the null hypothesis

  • probability of getting a test statistic at least as extreme as the one observed
    • assuming the null hypothesis is true
  • probablity of getting the observed value of the test statistic, or a value with at least as much evidence against the null hypothesis
  • assuming the null hypothesis is true
    • chances of seeing what we saw if null is true
  • the smaller the p-value, the greater the evidence against the null
    • if null is true, then the % of the p-value is the chance we have to find what we did
      • p-value very small = we'd have to be extremely (un)lucky to get the test we're using as evidence
  • if we are carrying out a test at the $\alpha$ level of significance, we can reject $H_0$ if p-value $\le \alpha$
    • hypothesis dont have confidence levels

interpreting p-values

• the smaller the p-value, the stronger the evidence against $H_0$
  • if we are carrying out the test at a fixed level of $\alpha$, the evidence against $H_0$ is statistically significant if p-value $\le \alpha$
  • if we get a big p-value, we still cannot state that the $H_0$ is true
  • distribution of the p-value
    • z/t test with assumptions being true
    • if $H_0$ is true
      • 
    • if $H_0$ is false
      • chance that we get a p-value of 1 (would be exactly on the mean) = that little bar
      • 
    • rough guideline
      • 

finding p-values

• example
  • suppose in a Z/t test the observed value of the test statistic is found to be 2.00
    • $H_0:\mu ={\mu }_0$, $H_a:\mu >{\mu }_0$
      • p-value = 1-pnorm(2) = 0.0228
    • $H_a:\mu <{\mu }_0$
      • pvalue = pnorm(2) = 0.977
    • $H_a:\mu \ne {\mu }_0$
      • p-value = 2* (1-pnorm(2)) = 0.0456