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14.7: Type II Error and Statistical Power

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In the prior example, the statistician failed to reject the Null Hypothesis because the probability of making Type I error (rejecting a true Null Hypothesis) exceeded the significance level of 5%. However, the statistician could have made Type II error if the machine is really operating improperly. One of the important and often overlooked tasks is to analyze the probability of making Type II error (β). Usually statisticians look at statistical power which is the complement of β.

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Beta (β): The probability of failing to reject the null hypothesis when it is actually false.

Power (or Statistical Power): The probability of rejecting  the null hypothesis when it is actually false.

Both beta and power are calculated for specific possible values of the Alternative Hypothesis.

  Fail to Reject Ho Reject Ho
Ho is true 1α α Type I error
Ho is false β Type II error 1β Power

If a hypothesis test has low power, then it would be difficult to reject Ho, even if Ho were false; the research would be a waste of time and money. However, analyzing power is difficult in that there are many values of the population parameter that support Ha. For example, in the soy sauce bottling example, the Alternative Hypothesis was that the mean was not 16 ounces. This means the machine could be filling the bottles with a mean of 16.0001 ounces, making Ha technically true. So when analyzing power and Type II error, we need to choose a value for the population mean under the Alternative Hypothesis (μa) that is “practically different” from the mean under the Null Hypothesis (μo). This practical difference is called the effect size.

Definition: Effect size

Effect Size: The “practical difference” between μo and μa=|μoμa|

where 

μo: The value of the population mean under the Null Hypothesis

μa: The value of the population mean under the Alternative Hypothesis

Suppose we are conducting a one‐tailed test of the population mean:  

Ho:μ=μ0Ha:μ>μ0

Consider the two graphs shown below. The top graph is the distribution of the sample mean under the Null Hypothesis, which was covered in an earlier section. The area to the right of the critical value is the rejection region.

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We now add the bottom graph, which represents the distribution of the sample mean under the Alternative Hypothesis for the specific value μa.

We can now measure the Power of the test (the area in green) and beta (the area in purple) on the lower graph.  

There are several methods of increasing Power, but they all have trade‐offs:

Ways to Increase Power Trade off
Increase Sample Size Increased cost or unavailability of data
Increase Significance level (α) More like to Reject a true Ho (Type I error)
Choose a value of μa further from μo Result may be less meaningful
Redefine population to lower standard deviation Result may be too limited to have value
Conduct as a one‐tail rather than a two‐tail test May produce a biased result

Example: Bus brake pads

Bus brake pads are claimed to last on average at least 60,000 miles and the company wants to test this claim. The bus company considers a “practical” value for purposes of bus safety to be that the pads last at least 58,000 miles. If the standard deviation is 5,000 and the sample size is 50, find the power of the test when the mean is really 58,000 miles. (Assume α=.05

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Solution

First, find the critical value of the test.

Reject Ho when Z<1.645

Next,  find the value of that corresponds to the critical value.

¯X=μo+Zσn=60000(1.645)(5000)/50=58837

Ho is rejected when ¯X<58837

Finally, find the probability of rejecting Ho if Ha is true.

P(¯X<58837)=P(Z<(58837μa)σ/n)=P(Z<(5883758000)5000/50)=P(Z<1.18)=.8810

Therefore, this test has 88% power and β would be 12%

Power Calculation Values

Input Values

μo = 60,000 miles

μa = 58,000 miles

α = 0.05 ݊

n = 50

σ = 5000 miles

Calculated Values

Effect Size = 2000 miles

Critical Value = 58,837 miles

β = 0.1190 or about 12%

Power = 0.8810 or about 88%

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This page titled 14.7: Type II Error and Statistical Power is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Maurice A. Geraghty via source content that was edited to the style and standards of the LibreTexts platform.

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