Identifying Type I and Type II errors

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to identify Type I and Type II errors.  Here's our problem statement: Identify the Type I error and the Type II error that correspond to the given hypothesis: The percentage of households with Internet access is less than 60%.

OK, in order to identify these errors, we need to know what they are, and to that end I prepared a small presentation to help us understand what they are.  With every null hypothesis that we have, the null hypothesis could be true or could be false.  We also have two actions that we can take with a null hypothesis; we can either reject the null hypothesis or fail to reject the null hypothesis.

In the event that the null hypothesis is actually false, we want to reject it.  We do this because we want to accept the null hypothesis only if it is true.  So when the null hypothesis is actually false, we want to reject the null hypothesis.  And when we do, we make a good conclusion.

However, if the null hypothesis is actually true and we reject it, well now we've done something we don't want to do.  We want to accept the null hypothesis if it's actually true, but if we end up rejecting it, that leads to what we call an error.  And specifically when we reject the null hypothesis when it's actually true, this is what we call a Type I error.

On the flip side of the coin, if we're going to fail to reject the null hypothesis, we want to do that when the null hypothesis is actually true.  So when we do, that leads to a good conclusion.  However, if we fail to reject the null hypothesis when it's false, that is another error, and that's what we call a Type II error — failing to reject a false null hypothesis.

Part 1

Let's bring this back to the problems that we have at hand.  We're first asked to identify the Type I error.  What I like to do when I look at these types of problems is identify what is the null hypothesis.  The hypothesis that we're given in the problem statement is not the null hypothesis; it is the alternative hypothesis.

We know this is true because by definition the null hypothesis is a statement of equality, and this statement that we have in the problem statement says “the percentage of households with Internet access is less than 60%.”  This is not a statement of equality.  Look at the words here — “is less than.”  This must then be the alternative hypothesis.  To create the null hypothesis, we simply make this statement one of equality.  So the percentage of households with Internet access is 60%; this is our null hypothesis.

The next thing I do is I remind myself of what the definition is of the error that I'm looking for.  So Type I error means we're rejecting the null hypothesis when the null hypothesis is actually true.  I write this definition out because what I'm going to do next is substitute in just like it were a mathematical equation this null hypothesis that we've identified into the definition.  So everywhere we see “null hypothesis,” I'm going to replace that with “percentage of households with Internet access equals 60%.”  This gives me what you see here.  “The Type I error is rejecting that the percentage of households with Internet access is 60% when the percentage of households with Internet access is 60%.”

Now I look at my answer options to determine which of these most closely matches what I've produced here.  And doing that, the Type I error is going to be rejecting the null hypothesis, so it's going to be answer option A or C.  “When it's actually equal to 60%” — that means we're looking at answer option A.  I select that option and check my answer.  Good job!

Part 2

And now the second part asks us to identify the Type II error.  To do that, I'm going to go back and repeat the same process that I used with the Type I error.  So I write out the definition for a Type II error.  The Type II error is failing to reject the null hypothesis when the null hypothesis is actually false.  I then make the substitution as I did before, and I get “the Type II error means failing to reject that the percentage of households with Internet access is 60% when the percentage of households with Internet access is less than 60%.”

Now I look at my answer options and see which ones most closely match what I’ve ended up written.  Here again it looks like it’s options A and C because we want to fail to reject the null hypothesis when that null hypothesis is actually false, meaning we're going to look at the alternative hypothesis of less than 60%.  Again, that's answer option A.  I check my answer.  Nice work!

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