Type 1 and type 2 errors

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Intros
Lessons
  1. What are type 1 and type 2 errors and how are they significant?
  2. Calculating the Probability of Committing a Type 1 Error
  3. Calculating the Probability of Committing a Type 2 Error
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Examples
Lessons
  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:

    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)

    1. "An artificial heart valve is malfunctioning"
    2. "A toy factory is producing defective toys"
    3. "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 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?
    1. 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.
      1. With a significance level of α\alpha=0.01 what can we say about Pacus' claim?
      2. 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?