Exam Tip: You will not need to know how to actually calculate P(Type II error ) or the power of the
test on the AP exam. You will need to understand the concept of each, what affects each, and how they
are related.example: Sticky Fingers is arrested for shoplifting. The judge, in her instructions to the jury, says
that Sticky is innocent until proven guilty. That is, the jury’s hypothesis is that Sticky is
innocent. Identify Type I and Type II errors in this situation and explain the consequence of
each.
solution: Our hypothesis is that Sticky is innocent. A Type I error involves mistakenly rejecting a
true hypothesis. In this case, Sticky is innocent, but because we reject innocence, he is found
guilty. The risk in a Type I error is that Sticky goes to jail for a crime he didn’t commit.
A Type II error involves failing to reject a false hypothesis. If the hypothesis is false, then Sticky is
guilty, but because we think he’s innocent, we acquit him. The risk in Type II error is that Sticky
goes free even though he is guilty.
In life we often have to choose between possible errors. In the example above, the choice was
between sending an innocent person to jail (a Type I error) or setting a criminal free (a Type II error).
Which of these is the most serious error is not a statistical question—it’s a social one.
We can decrease the chance of Type I error by adjusting α. By making a very small, we could virtually
ensure that we would never mistakenly reject a true hypothesis. However, this would result in a large
Type II error because we are making it hard to reject the null under any circumstance, even when it is
false.
The probability of making a Type II error is smaller and, hence, the power of the test greater if:
• The sample size is increased.
• The standard deviation of our sample data is decreased (this is not always under the control of the
researcher but, for example, if more precise measurements are possible the variability in the data could
be reduced.).
• We increase the significance level (α). (This could be seen as dishonest – manipulating the significance
level to get the result you want.)
• The effect size (the difference between the hypothesized parameter and the true value) is larger. A
bigger difference is easier to detect!
example: A package delivery company claims that it is on time 90% of the time. Some of its
clients aren’t so sure, thinking that there are often delays in delivery beyond the time promised.
The company states that it will change its delivery procedures if it is wrong in its claim.
Suppose that, in fact, there are more delays than claimed by the company. Which of the
following is equivalent to the power of the test?
(a) The probability that the company will not change its delivery procedures
(b) The P -value > α
(c) The probability that the clients are wrong