Type 1 and type 2 errors

Type 1 and type 2 errors

Lessons

Type 1 Errors:

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$)=1-P($Fail to Reject $| H_0$ is false$)=1-\beta$

Recall:
Test Statistic:
Proportion:
$Z=\frac{\hat{p}-p}{\sqrt{\frac{p(1-p)}{n}}}$

Mean:
$Z=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$
• Introduction
a)
What are type 1 and type 2 errors and how are they significant?

b)
Calculating the Probability of Committing a Type 1 Error

c)
Calculating the Probability of Committing a Type 2 Error

• 1.
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)
a)
"An artificial heart valve is malfunctioning"

b)
"A toy factory is producing defective toys"

c)
"A newly designed car is safe to drive"

• 2.
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?

• 3.
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. a) With a significance level of $\alpha$=0.01 what can we say about Pacus' claim? b) Unbeknownst to me the actual average salary of a teacher is$61,000. What is the probability of committing a type 2 error when testing Pacus' claim?