Transcript AP Statistics

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 𝐻0 when 𝐻0 is
true. 𝑃 𝑟𝑒𝑗𝑒𝑐𝑡 𝐻0 𝐻0 𝑖𝑠 𝑡𝑟𝑢𝑒) = 𝛼
Type II error: The probability that we accept 𝐻0 when 𝐻𝑎 is
true. 𝑃 𝑎𝑐𝑐𝑒𝑝𝑡 𝐻0 𝐻𝑎 𝑖𝑠 𝑡𝑟𝑢𝑒) = 𝛽
Power: The probability that we reject 𝐻0 when 𝐻𝑎 is true.
𝑃 𝑟𝑒𝑗𝑒𝑐𝑡𝐻0 𝐻𝑎 𝑖𝑠 𝑡𝑟𝑢𝑒) = 1 − 𝛽
• Points regarding Power:
1. Be sure you can interpret in the context of a problem.
2. The standard power level is 0.80.
3. Tests usual report power instead of type II error.
4. The higher the power the more sensitive the test is to
specific alternatives.
Diagram:
Decision
based on a Reject Ho
Accept Ho
sample.
Truth about the Population
Ho is true
Ha is true
• 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 𝑯𝒂 : 𝝁 > 𝝁𝟎 .
• Ex 3. Draw two normal curves and label Type I error,
Type II error and Power. Use 𝑯𝒂 : 𝝁 < 𝝁𝟎 .
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 = 𝜇𝑎 ). 𝜇𝑎 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: 𝐻0 : 𝜇 = 8; 𝐻𝑎 : 𝜇 < 8; 𝜎 = 0.5; n=25; 𝜇𝑎 = 7.75