Section 11.4: Using Inference to Make Decisions

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Section 11.4: Using Inference to Make Decisions AP Statistics Section 11.4: Using Inference to Make Decisions

Objective: To be able to calculate and understand Type I error, Type II error and Power. Type I error: The probability that we reject 𝐻 0 when 𝐻 0 is true. 𝑃 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻 0 𝐻 0 𝑖𝑠 𝑡𝑟𝑢𝑒)=𝛼 Type II error: The probability that we accept 𝐻 0 when 𝐻 𝑎 is true. 𝑃 𝑎𝑐𝑐𝑒𝑝𝑡 𝐻 0 𝐻 𝑎 𝑖𝑠 𝑡𝑟𝑢𝑒)=𝛽 Power: The probability that we reject 𝐻 0 when 𝐻 𝑎 is true. 𝑃 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻 0 𝐻 𝑎 𝑖𝑠 𝑡𝑟𝑢𝑒)=1−𝛽

Points regarding Power: Be sure you can interpret in the context of a problem. The standard power level is 0.80. Tests usual report power instead of type II error. The higher the power the more sensitive the test is to specific alternatives. Diagram: Truth about the Population Decision based on a sample. Ho is true Ha is true Reject Ho Accept Ho

Ex 1. “Innocent until proven guilty.” State the hypotheses in terms of the expression. Interpret Type I and Type II errors in the context of the expression. Type I error: Type II error: Which is more costly and why?

Ex 2. Draw two normal curves and label Type I error, Type II error and Power. Use 𝑯 𝒂 : 𝝁> 𝝁 𝟎 .

Ways to increase power: 1. 2. 3. 4 Ways to increase power: 1. 2. 3. 4. Method for calculating Type II error: 1. Write a rule for accepting Ho in terms of 𝑥 . 2. Find P(accept Ho | 𝜇= 𝜇 𝑎 ). 𝜇 𝑎 is a value that is provided to us in the region of Ha.

Ex 4. Calculate Type II error and power given the following scenario: 𝐻 0 :𝜇=8; 𝐻 𝑎 :𝜇<8; 𝜎=0.5; n=25; 𝜇 𝑎 =7.75