A statistical hypothesis is a statement about the population/its distribution

A statement that fully specifies the distribution of a population is called a simple hypothesis. Otherwise, the statement is called a composite hypothesis

• one set value vs a range of multiple values for $\mu$ for example

Null Hypothesis, $H_0:$ disapproves the claim
Alternative Hypothesis, $H_a:$ claim under investigation/research

where parameter space $\mathrm{\Theta }={\mathrm{\Theta }}_0\cup {\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_0\cap {\mathrm{\Theta }}_1=\mathrm{\varnothing }$

• only one of the hypothesis can be accepted

Error types

Examples

• sleep study, total sleep recorded and subjective time they thought they slept
  • ${\mu }_I$ = mean of the difference in the answer x reality for insoniacs
  • ${\mu }_N$ = mean of the difference in the answer x reality for normal people
  • $H_0:{\mu }_I={\mu }_N$
  • $H_a:{\mu }_I\ne {\mu }_N$
  • on average, do normal sleepers perceive sleep time correctly?
  • does the mean mismatch time
• hand grip strenght x attractiveness
  • $H_0$ : there is no relationship
  • $H_a$ : there is a relationship
  • we might assume that if there is a relationship, it is a linear one, and test
    • $H_0$ : true slope is 0
    • $H_a$ : true slope is not 0
• company's arsenic level
  • government trying to prove that it goes over 0.30
  • $H_0:\mu =0.30$
    • $H_0:\mu \le 0.30$ makes more sense, but in the end we are only gonna pick one value when proving it with formulas, and thats just gonna be the closest value to the $H_a$ (0.30)
  • $H_a:\mu >0.30$