Question

Explain the concept of statistical power. Briefly explain how power is calculated and what information you might have to assume or know in order to make this calculation.

Answer 1

The power of any test of statistical significance is defined as the probability that it will reject a false null hypothesis. Statistical power is inversely related to betaor the probability of making a Type II error. In short, power = 1 – β.

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 lower the significance level, the lower the power of the test. If you reduce the significance level (e.g., from 0.05 to 0.01), theregion of acceptance gets bigger. As a result, you are less likely to reject the null hypothesis. This means you are less likely to reject the null hypothesis when it is false, so you are more likely to make a Type II error. In short, the power of the test is reduced when you reduce the significance level; and vice versa.
• 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.

Suppose a child psychologist says that the average time that working mothers spend talking to their children is 11 minutes per day. You want to test

versus

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4. FINDING THE POWER OF A HYPOTHESIS TEST

FINDING THE POWER OF A HYPOTHESIS TEST

RELATED BOOK

U Can: Statistics For Dummies

By Deborah J. Rumsey, David Unger

When you make a decision in a hypothesis test, there’s never a 100 percent guarantee you’re right. You must be cautious of Type I errors (rejecting a true claim) and Type II errors (failing to reject a false claim). Instead, you hope that your procedures and data are good enough to properly reject a false claim.

The probability of correctly rejecting H0when it is false is known as the power of the test. The larger it is, the better.

Suppose you want to calculate the power of a hypothesis test on a population mean when the standard deviation is known. Before calculating the power of a test, you need the following:

• The previously claimed value of

  

  in the null hypothesis,

  

• The one-sided inequality of the alternative hypothesis (either < or >), for example,

  

• The mean of the observed values

  

• The population standard deviation

  

• The sample size (denoted n)

• The level of significance

  

To calculate power, you basically work two problems back-to-back. First, find a percentile assuming that H0 is true. Then, turn it around and find the probability that you’d get that value assuming H0 is false (and instead Ha is true).

1. Assume that H0 is true, and

   

2. Find the percentile value corresponding to

   

   sitting in the tail(s) corresponding to Ha. That is, if

   

   then find b where

   

   If

   

   then find b where

   

3. Assume that H0 is false, and instead Ha is true. Since

   

   under this assumption, then let

   

   in the next step.

4. Find the power by calculating the probability of getting a value more extreme than b from Step 2 in the direction of Ha. This process is similar to finding the p-value in a test of a single population mean, but instead of using

   

   you use

   

Suppose a child psychologist says that the average time that working mothers spend talking to their children is 11 minutes per day. You want to test

versus

You conduct a random sample of 100 working mothers and find they spend an average of 11.5 minutes per day talking with their children. Assume prior research suggests the population standard deviation is 2.3 minutes.

When conducting this hypothesis test for a population mean, you find that the p-value = 0.015, and with a level of significance of

you reject the null hypothesis. But there are a lot of different values of

(not just 11.5) that would lead you to reject H0. So how strong is this specific test? Find the power.

1. Assume that H0 is true, and

   

2. Find the percentile value corresponding to

   

   sitting in the upper tail. If p(Z > zb) = 0.05, then zb = 1.645. Further,

   

3. Assume that H0 is false, and instead

   

4. Find the power by calculating the probability of getting a value more extreme than b from Step 2 in the direction of Ha. Here, you need to find p(Z> z) where

   

   Using the Z-table, you find that