AP STATISTICS LESSON 10 – 4 (DAY 2) POWER

ESSENTIAL QUESTION: What is power and how is it determined in a type II error? Objectives: To define power for a type II error. To calculate and increase the power.

Power A test makes a Type II error when it fails to reject a null hypothesis that really is false. A high probability of a Type II error for a particular alternative means that the test is not sensitive enough to usually detect that alternative. The power of a test against any alternative is 1 minus the probability of a Type II error for that alternative.

Example 10.23 Page 600 Exercise is Good We will answer this question by calculating the power of the significance test that will be used to evaluate the data to be collected. The calculations consist of three steps. Step 1: State H0, Ha, the particular alternative we want to detect, and the significance level α. Step 2: Find the values of x that will lead to reject Ho. Step 3: Calculate the probability of observing these values of x when the alternative is true.

High Power is Desirable Along with 95% confidence intervals and 5% significance tests, 80% power is becoming a standard. Many U.S. government agencies that provide research funds require that the sample size for the funded studies be sufficient to detect important results 80% of the time using a 5% test of significance.

Increasing the Power Suppose you have performed a power calculation and found that the power is too small. What can you do to increase it? Increase α. A 5% test of significance will have a greater chance of rejecting the alternative than a 1% test because the strength of evidence required for rejection is less. Consider a particular alternative that is farther away from μo. Values of μ that are in Ha but lie close to the hypothesized value μo are harder to detect (lower power) than values of μ that are far from μo.

Increasing the Power (continued…) Increase the sample size. More data will provide more information about x so we have a better chance of distinguishing values of μ. Decrease σ. This has the same effect as increasing the sample size more information about μ. Improving the measurement process and restricting attention to a subpopulation are two common ways to decrease σ. A null hypothesis that is in fact false can become widely believed if repeated attempts to find evidence against it fail because of low power.

Testing Hypotheses This method mixes the reasoning of significance tests and decision rules a follows: State Ho and Ha just as in a test of significance. In particular, we are seeking evidence against Ho. Think of the problem as a decision problem, so that the probabilities of type I and type II errors are relevant. Because of Step 1, type I errors are more serious. So choose an α (significance level) and consider only tests with probability of type I error no greater than α. Among these tests, select one that makes the probability of a type II error as small as possible (that is, power as large as possible). If this probability is too large, you will have to take a larger sample to reduce the chance of an error.