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Chapter 10

Chapter 10. Section 10.4 Part 1 – Inference as Decision. Inference as Decision. Tests of significance assess the strength of evidence against the null hypothesis.

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Chapter 10

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  1. Chapter 10 Section 10.4 Part 1 – Inference as Decision

  2. Inference as Decision • Tests of significance assess the strength of evidence against the null hypothesis. • The alternative hypothesis (the statement we seek evidence for) enters the test only to help us see what outcomes count against the null hypothesis. • Using significance tests with fixed, however, suggests another way of thinking. • A level of significance chosen in advance points to the outcome of the test as a decision. • The transformation from measuring the strength of evidence to making decisions is not a small step.

  3. Acceptance Sampling • There are circumstances that call for a decision or action as the end result of inference. • Acceptance samplingis one such circumstance. • We will use acceptance sampling to show how a different concept – inference as decision – changes the reasoning used in tests of significance.

  4. Type I and Type II Errors • There are simply two hypotheses, and we must accept one and reject the other. • It is convenient to continue to call the two hypotheses Ho and Ha , but Ho no longer has the special status (the statement we try to find evidence against) that it had in tests of significance. • In the acceptance sampling problem, we must decide between Ho and Ha.

  5. Type I and Type II Errors (continued…) • If we reject Ho (accept Ha) when in fact Ho is true, this is a Type I error. • If we accept (reject Ha ) H0 when in fact Ha is true, this is a Type II error.

  6. The Two Types of Error in Testing Hypotheses

  7. Error Probabilities • We assess any rule for making decisions by looking at the probabilities of the two types of errors. • This is in keeping with the idea that statistical inference is based on asking, “ What would happen if I used this procedure manytimes?” • Significance tests with fixed level give a rule for making decisions, because the test either rejects Ho or fails to reject it. • We then describe the performance of a test by the probabilities of type I and type II errors.

  8. Example 10.21 – Are These chips Too Salty? • Read the scenario on p.595 • Steps for finding type I and type II errors: • Step 1: To find the type I error you use the or significance level = type I error. • Finding type II error: • Step 2: find Z values for level: Ho : μ = 2 Ha : μ ≠ 2

  9. The light shaded areais a Type 1 error (the probability of rejecting Ho: μ = 2 when in fact μ = 2. • The probability of a Type II error (dark shaded area) is the probability of accepting Ho when in fact μ = 2.05.

  10. Significance and Type I Error • The significance level α of any fixed level test is the probability of a Type I error. • That is, is the probability that the test will reject the null hypothesis Ho when Ho is in fact true.

  11. Example 10.21 (continued…) • Step1: write the rule in terms of . 1.9723 ≤ ≤ 2.027 • Step 2: find the probability of accepting Ho assuming that the alternative is true. Take μ = 2.05 and standardize to find the probability.

  12. Interpreting Type II Error The probability of a type II error in example 10.22 is .0571. This tells us that this test will lead us to fail to reject Ho : μ = 2 for about 6% of all batches of chips with a μ = 2.05. In other words, we will accept 6% of batches of potato chips so bad that their mean salt content is 2.05 mg.

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