Lesson 9.5 The Type I and Type II Errors

Type I Error

A Type I error occurs when we reject H_o when H_o is really true.

Fixed or pre-conceived α (the same α we used in hypothesis testing) is the probability of rejecting H_o when H_o is really true. So,

We always try to minimize α.

Example: Suppose H_o is

The Type I error is

We do not think the coin is fair when, in fact, it really is.

So, α = P(Type I error) = the probability that we think the coin is not fair when, in fact, it really is.

Example: Suppose H_o is

where p is the proportion of Americans who vote in presidential elections. In words, H_o is "At most, 60% of Americans vote in presidential elections."

The Type I error is

We think that the proportion of Americans who vote in presidential elections is more than 60% when, in fact, the proportion is at most 60%.

(If we reject H_o, then we think that p > 0.60.)

So, α = P(Type I error) = the probability that we think the proportion of Americans who vote in presidential elections is more than 60% when, in fact, the proportion is at most 60%.

Example: Suppose H_o is

where μ is the average cost, in dollars, of dinner at the better restaurants in Silicon Valley.

The Type I error is:

We think that the average cost of dinner at the better restaurants in Silicon Valley is less than 25 dollars when, in fact, the cost is at least 25 dollars.

(If we reject H_o, then we think that the average cost is less than 25 dollars.)

So, α = P(Type I error) = the probability that we think the average cost of dinner at the better restaurants in Silicon Valley is less than 25 dollars when, in fact, the cost is at least 25 dollars.

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Type II Error

A Type II error occurs when we do not reject H_o when H_o is really false.

We use the Greek letter β as the probability of not rejecting H_o when H_o is really false. So,

β = P(Type II error).

We always try to minimize b.

Example: Suppose H_o is

The coin is fair.

The Type II error is

We think the coin is fair when, in fact, it really is not fair.

So, β = P(Type II error) = the probability that we think the coin is fair when, in fact, it really is not fair.

Example: Suppose H_o is

where p is the proportion of Americans who vote in presidential elections. In words, H_o is "At most, 60% of Americans vote in presidential elections."

The Type II error is

We think that the proportion of Americans who vote in presidential elections is at most 60% when, in fact, the proportion is more than 60%.

 So, β = P(Type II error) = the probability that we think the proportion of Americans who vote in presidential elections is at most 60% when, in fact, the proportion is more that 60%.

Example: Suppose H_o is

where μ is the average cost, in dollars, of dinner at the better restaurants in Silicon Valley.

The Type II error is:

We think that the average cost of dinner at the better restaurants in Silicon Valley is at least 25 dollars when, in fact, the cost is less than 25 dollars.

 So, β = P(Type II error) = the probability that the average cost of dinner at the better restaurants in Silicon Valley is at least 25 dollars when, in fact, the cost is less than 25 dollars.

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Decision Table

We can summarize the Type I and Type II errors and the correct decisions in a table.

The Power of the Test is defined to be 1 - β.

1 - β is the probability of rejecting H_o when, in fact, H_o is really false. It is desirable to have a high power since we would always want to reject H_o when H_o is really false.

Besides being used in the Power of the Test, β is also used to determine the appropriate sample size for the hypothesis test. You do not study β extensively in this course. If you were to take the next statistics course, you would learn more about β.

Think About It

To help you better understand the Type I and Type II errors, do the Try-It examples Introductory Statistics. You can check some of your answers in the back of the book.

This is the last section of this lesson.

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Up » 9.1 Hypothesis Testing » 9.2 Hypothesis Testing - Known » 9.3 Hypothesis Testing- Unknown » 9.4 Hypothesis Testing for a Single Population Proportion » 9.5 Type I and II Errors