### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

### Beyond AP Statistics

#### Probability Basics

#### Small Samples

#### Distributions

#### Power

# Power of a Hypothesis Test

The probability of *not* committing a
Type II error is called the **power** of
a hypothesis test.

## Effect Size

To compute the power of the test, one offers an alternative
view about the "true" value of the population parameter,
assuming that the null hypothesis is false. The
**effect size** is the difference
between the true value and the value specified in the null
hypothesis.

Effect size = True value - Hypothesized value

For example, suppose the null hypothesis states that a population mean is equal to 100. A researcher might ask: What is the probability of rejecting the null hypothesis if the true population mean is equal to 90? In this example, the effect size would be 90 - 100, which equals -10.

## Factors That Affect Power

The power of a hypothesis test is affected by three factors.

- Sample size (
*n*). Other things being equal, the greater the sample size, the greater the power of the test. - Significance level (α). The smaller the significance level, the higher the power of the test. If you make the significance level smaller (e.g., from 0.05 to 0.01), you reduce the region of acceptance. As a result, you are more likely to reject the null hypothesis. This means you are less likely to accept the null hypothesis when it is false; i.e., less likely to make a Type II error. Hence, the power of the test is increased.
- The "true" value of the parameter being tested. The greater the difference between the "true" value of a parameter and the value specified in the null hypothesis, the greater the power of the test. That is, the greater the effect size, the greater the power of the test.

## Test Your Understanding

**Problem 1**

Other things being equal, which of the following actions will reduce the power of a hypothesis test?

I. Increasing sample size.

II. Reducing significance level (e.g., from 0.05 to 0.01).

III. Increasing beta, the probability of a Type II error.

(A) I only

(B) II only

(C) III only

(D) All of the above

(E) None of the above

**Solution**

The correct answer is (C). Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Making the significance level smaller reduces the region of acceptance, which makes the hypothesis test more likely to reject the null hypothesis, thus increasing the power of the test. Since, by definition, power is equal to one minus beta, the power of a test will get smaller as beta gets bigger.

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**Problem 2**

Suppose a researcher conducts an experiment to test a hypothesis. If she doubles her sample size, which of the following will increase?

I. The power of the hypothesis test.

II. The effect size of the hypothesis test.

III. The probability of making a Type II error.

(A) I only

(B) II only

(C) III only

(D) All of the above

(E) None of the above

**Solution**

The correct answer is (A). Increasing sample size makes the hypothesis test more sensitive - more likely to reject the null hypothesis when it is, in fact, false. Thus, it increases the power of the test. The effect size is not affected by sample size. And the probability of making a Type II error gets smaller, not bigger, as sample size increases.

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