Quants Summary

Hypothesis Testing

• Difference between sample statistic and population parameter
  • number that describes
  • variable (fixed), known (unknown)
  • common statistics: mean, SD, proportion, size
• What is population (in research)? …and sample?
• General idea:
  • initial assumption ($H_0$)
  • collect evidence
  • based on available data, decide whether to accept/reject ($H_0$)
• Null ($H_0$): no relationship / is different
• Alternative ($H_1$): used in stat. inference contrary to $H_0$

Types of Error	$H_0$ is false	$H_0$ is true
Reject $H_0$	correct	Type $I$
Accept $H_0$	Type $II$	correct

$$P(\text{Type I error})=\alpha$$ $$P(\text{Type II error})=\beta$$

P(reject $H_0$ when not true) = $1-\alpha$ P(accept $H_0$ when its true) = $1-\beta$ (Power of the test)

• Higher the power lower the probability of Type $II$ error.

• Power to accept the right thing when its true.

• Type $I$ = Producer’s risk, “I, as a producer rejected a good saying that it was faulty, when it was actually not. Loss for the producer.”

• Type $II$ = Consumer’s risk, “I, the consumer got a faulty product because the producer said that it was okay, when it was actually faulty”

Steps

1. frame $H_0$ and $H_1$
2. collect data (sampling, appropriate, good representation)
3. perform statistical test (on the basis of data)
   • $t$-test, ${\chi }^2$-test, $z$-test, ANOVA test
4. Find the table value for the given significance level $\alpha$ from the table.
5. Draw inferences based on the calculated and tabulated value of statistic.
   • If $z_{\text{calc}}>z_{\text{tab}}$ : reject $H_0$
   • If $z_{\text{calc}}<z_{\text{tab}}$ : don’t reject $H_0$