Hypothesis Tests

Chapter 7

Hypothesis Tests:

A test of competing hypothesis.

• $H_o$ : Null hypothesis
  • Default conditions
  • Can never be proved. The test can never prove the null
• $H_a$ : Alternative Hypothesis
  • What you are trying to prove

Examples:

1. Legal Justice System; Innocent until proven guilty (Innocent = $H_O$), (Guilty = $H_a$).

   • If the evidence in far enough away from innocence, you are proven to be guilty at some level of confidence. Here, if you make a mistake in your alternative hypothesis, somebody goes to jail innocently

2. The Scientific Method

   • $H_o$ = Current Theory (say the theory of general relativity right now).
   • $H_a$ = Your new theory
     • So you gather data, and if it sufficiently supports your theory, (i.e, far enough away from $H_o$ in the direction of $H_a$)

---

Example 1

Note: The equality must always be on $H_o$. We test ${\mu }_x=10$ (hardest to reject, and gives us default mean)

If so, we reject $H_o$ and conclude, with at least ($1-\alpha$) confidence, that $H_a$ is correct.

Note: This is a one-sided test. (when we have a less than, or greater than $H_a$). We’d use $Z_{\alpha }$ or $T_{\alpha }$

---

Example 2 (Two sided test this time)

(Two sided. So we use $Z_{\frac{\alpha }{2}})$

---

Types of Errors

• Type 1: Rejecting $H_o$ when it is true

  • P[Type 1 Error] = $\alpha$
    • We choose this so choose $\alpha$ to be small, but not too small. $\alpha$ of 0.05 or 0.10 is selected

• Type 2: Failing to reject $H_o$ when it is false

  • Problem: If $H_o$ is false, we do not know what ${\mu }_x$ is. Our $H_o$ tells us the mean by default, so that being wrong is a big problem
    • P[Type 2 Error] = $\beta$ $\leftarrow$ (harder to computer, so we’ll basically ignore this)
      • This increases as $\alpha$ decreases (again, why we don’t want to choose $\alpha$ too small)

---

P - Value

This is the value of $\alpha$ at the boundary of rejecting $H_o$

= The probability of observing a future test statistic at least as extreme as the one observed in the direction of $H_a$

---

Example

Suppose $H_o$ : ${\mu }_x=$ 0 $H_o$ : ${\mu }_x\ne$ 20

and we have computed a test statistic

$$Z=\frac{()-()}{()}=2.6$$

(this means that our $\bar{X}$ *is 2.6 standard deviations away from $H_o$ : ${\mu }_x=$ 20)

P-value (i.e. = $P[Z>2.6\cup Z<-2.6]$ = 2($1-\varphi (2.6))$

Small P-values support $H_a\to$ suggests that $H_o$ is false, and should be rejected. If $\alpha$ is selected to be greater than the P-value, we reject $H_o$

Large P-Values support $H_o$

---