Type 1 and type 2 errors
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Lessons
 Determining the Significance of Type 1 and Type 2 Errors
What are the Type 1 and Type 2 Errors of the following null hypotheses :
This table may be useful:
$H_0$ is true
$H_0$ is false
Reject $H_0$
Type 1 Error (False Positive)
Correct Judgment
Fail to Reject $H_0$
Correct Judgment
Type 2 Error (False Negative)
 Calculating the probability of Committing Type 1 and Type 2 Errors
Suppose 8 independent hypothesis tests of the form $H_0:p=0.75$ and $H_1:p$ < $0.75$ were administered. Each test has a sample of 55 people and has a significance level of $\alpha$=0.025. What is the probability of incorrectly rejecting a true $H_0$ in at least one of the 8 tests?  Pacus claims that teachers make on average less than $66,000 a year. I collect a sample of 75 teachers and find that their sample average salary is $62,000 a year. The population standard deviation for a teacher's salary is $10,000 a year.
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Topic Notes
A type 1 error is the probability of rejecting a true $H_0$.
$\alpha=P($reject $H_0$True $H_0)$
So in this case our hypothesis test will reject what is a true $H_0$.
Type 2 Errors:
A type 2 error is the probability of failing to reject a false $H_0$.
$\beta=P($Failing to Reject $H_0$False $H_0)$

$H_0$ is true 
$H_0$ is false 
Reject $H_0$ 
Type 1 Error (False Positive) 
Correct Judgment 
Fail to Reject $H_0$ 
Correct Judgment 
Type 2 Error (False Negative) 
The Power of a Hypothesis Test is the probability of rejecting $H_0$ when it is false. So,
Power $=P($Reject $ H_0$ is false$)=1P($Fail to Reject $ H_0$ is false$)=1\beta$
Recall:
Test Statistic:
Proportion: $Z=\frac{\hat{p}p}{\sqrt{\frac{p(1p)}{n}}}$
Mean: $Z=\frac{\overline{x}\mu}{\frac{\sigma}{\sqrt{n}}}$
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