Type I: $H_0$ is true.

$P(Type\ I)=P(rejecting H_0|H_0\ is\ true)$

Type II: $H_0$ is not true.

$P(Type\ II)=P(not\ rejecting H_0|H_0\ is\ false)$

considering the length of text, “II’ is longer than “I”, similarly “is not true” is longer than “is true”

• We can control the probability of Type I error easily through $\alpha$
• The probability of Type I and Type II are related. one ⬆️, another ⬇️

Usual strategy

Choosing a value for $\alpha$ so that the probability of a Type I error will be reasonably small.

Then one computes the probability of a Type II error, and hopes that it is not too large.

When obtaining $P(Type\ II\ error)$, we need to assume a “true” mean $\mu$ under $H_1$.

How to decrease $P(Type\ II\ error)$

1. increase sample size $n$
2. increase significance level $\alpha$

6.9.1 Power of Test

$$ power=1-P(Type\ II\ error) $$

Derivation

How to compute the power

1. Compute the rejection region
2. Compute the probability that the test statistic falls in the rejection region if $H_1$ is true.

In general, test with $power\gt 0.80$ is considered acceptable.

Example 13

Example 14 *