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 H0H_0.

α=P(\alpha=P(reject H0 H_0| True H0)H_0)

So in this case our hypothesis test will reject what is a true H0H_0.

Type 2 Errors:

A type 2 error is the probability of failing to reject a false H0H_0.

β=P(\beta=P(Failing to Reject H0H_0|False H0)H_0)

H0H_0 is true

H0H_0 is false

Reject H0H_0

Type 1 Error (False Positive)

Correct Judgment

Fail to Reject H0H_0

Correct Judgment

Type 2 Error (False Negative)



The Power of a Hypothesis Test is the probability of rejecting H0H_0 when it is false. So,
Power =P(=P(Reject H0| H_0 is false)=1P()=1-P(Fail to Reject H0| H_0 is false)=1β)=1-\beta

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

Mean:
Z=xμσnZ=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}
  • 1.
    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


  • 2.
    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:

    H0H_0 is true

    H0H_0 is false

    Reject H0H_0

    Type 1 Error (False Positive)

    Correct Judgment

    Fail to Reject H0H_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”


  • 3.
    Calculating the probability of Committing Type 1 and Type 2 Errors
    Suppose 8 independent hypothesis tests of the form H0:p=0.75H_0:p=0.75 and H1:pH_1:p < 0.750.75 were administered. Each test has a sample of 55 people and has a significance level of α\alpha=0.025. What is the probability of incorrectly rejecting a true H0H_0 in at least one of the 8 tests?

  • 4.
    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 α\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?