Type I and Type II Errors

For type I and type II errors, we must know the null and alternate hypotheses. H 0 : µ = 40 The mean of the population is 40. H a : µ ≠ 40 The mean of the population is not 40. For our example to follow: P(Type I error) = αP(Type II error) = β We can define:

Type I error occurs when the null hypothesis is true, but our sample mean is extreme, leading us to believe that the null hypothesis is false. We establish our planned amount of Type I error when we choose α. Of course, we expect to be correct in failing to reject the null hypothesis 1-α, when the null hypothesis is true.

When we choose a level of significance we set α. Our sampling distribution has a mean of 40 and standard deviation of 10, based on population standard deviation of 80 with a sample size of 64. The red regions represent regions of Type I error and have probability α. In this example α is 5%. We get a Type I error when our distribution is centered at 40, but our sample mean happens to be larger than 59.6 or smaller than 20.4.

If we change the significance level to 90%, we change α. Here α is 10%. As you can see, we have increased the probability of Type I error.

Type II error occurs only when the null hypothesis is false. It cannot occur if the null hypothesis is true, by our very definition. When we speak of Type II error we must know that the null hypothesis is not true.

Let’s start with our hypothetical distribution: Now we add an alternate distribution. Our sample mean will come from this sampling distribution, N(68,10), instead of the hypothetical sampling distribution.

The region shaded pink is our probability of Type I error, here 5%. The region shaded blue is the probability of Type II area. Notice that it is under the alternate (blue) distribution.

We make a Type II error whenever the null hypothesis is false, but we get a sample mean that falls into a range that will cause us to fail to reject the null hypothesis. Let’s take a closer look: Sample means between 20 and 60 will “look good” to us, we will not reject the null hypothesis.

Now we check the alternate distribution. Are there times when sample means from this distribution will give us values between 20 and 60? In fact, there are.

To find the probability of Type II area we find the area under the curve. So the probability of Type II error is 21%.

That is, when the true mean is 68, there is a 21% probability that we will fail to reject the null hypothesis. How can we reduce the probability of Type II error? Examine the following figures: Can you see that β is less now, but α is greater?

Here the probability of Type II error is 12.4%

Increasing α does result in a decrease in β. This does not necessarily get you very far ahead. Suppose we could compare to a different alternate distribution. Suppose we could make it have a larger mean, perhaps 72 instead of 68. Would this change β?

Now we have a new alternate distribution N(72,10) and so a new probability. So we now have 11.5% Type II error. While moving the alternate distribution further away reduces Type II error, we cannot always do this. Another approach is to decrease standard deviation. Any way we can accomplish this will have the same effect. Usually you can change sample size.

If our sampling distributions are now N(40,8) and N(68,8) we can find the effect on probability of Type II error. This also shows a reduction in Type II error. Increasing sample size will be our most effective way to minimize Type II error.

With a decrease in the standard deviation we see the probability of Type II error decrease to 6%. Decreasing the standard deviation reduces the amount of overlap between the two distributions, thus reducing the Type II error.

We have seen the difference between Type I and Type II errors. We set the probability of Type I error when we choose a level of significance. The probability of Type II error can be reduced by increasing α, by reducing the standard deviation (perhaps by increasing sample size), or by increasing the distance between the hypothetical and alternate means.

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