AP Statistics
I.
Inference Part 2
Type I & II Errors & Power of a Test
Unit 9 Day 4
Type I and Type II Errors
-Rejecting or failing to reject the null hypothesis is based on probability. Nothing that we do is a certain result,
so there is always the chance of making an error. Either we reject the null when we shouldn’t have, or we fail
to reject it when it really is false. These are called Type I and Type II errors.
Type I Error: reject the null hypothesis when it is actually true
Type II Error: failing to reject the null hypothesis when it is actually false. Remember: Fail “II” Reject the null when
it’s false.
-the probability of a Type I error is the probability or rejecting the null, when it is actually true. This is the same as
the alpha level for the test.
-P(Type I Error) = P(reject H0 | H0 is true ) – conditional probability
-as the probability of a Type I error decreases (lowering α), the probability of a Type II error will increase
II.
Power of a Test
The power of a test against a specific alternative is the probability that the test will reject H 0 at a chosen
significance level alpha when the specified alternative value of the parameter is true. This is basically the
probability of making a correct decision…higher power means a stronger test.
Example 1) A company has developed a new deluxe AAA battery that is supposed to last longer than its
regular AAA battery. However, these new batteries are more expensive to produce, so the company would
like to be convinced that they really do last longer. Based on years of experience, the company knows that its
regular AAA batteries last for 30 hours of continuous use, on average. The company selects an SRS of 15 new
batteries and uses them continuously until they are completely drained. A significance test is performed using
these hypotheses:
H0: µ = 30 hours
µ is the true mean lifetime of the new deluxe AAA batteries.
Ha: µ > 30 hours
-The resulting P-value is 0.0276.
a). What conclusion would you make for each of the following levels?
i. α = 0.05
ii. α = 0.01
-At the 0.05 level, I would reject the null hypothesis & conclude that the true mean lifetime of the new deluxe
AAA batteries is longer than 30 hours.
-At the 0.01 level, I would fail to reject the null hypothesis & could not conclude that the true mean life time of
the new deluxe AAA batteries is longer than 30 hours.
b). Describe Type I and Type II errors for this scenario in context.
-Type 1 error: Rejecting the null when it is true. This means I would conclude that the true mean lifetime of the
new AAA batteries is longer than 30 hours, when it actually isn’t. The company would waste money on
producing the batteries when they are no different than the cheaper ones.
-Type II error: Failing to reject the null when it is false. This means I would conclude that the new batteries are
no different than the older ones and miss out on an opportunity to sell more expensive batteries that last longer.
2. When a law firm represents a group of people in a class action lawsuit and wins that lawsuit, the firm
recieves a percentage of the group’s monetary settlement. That settlement amount is based on the total
number of people in the group – the larger the group and the larger the settlement, the more money the firm
will receive.
A law firm is trying to decide whether to represent car owners in a class action lawsuit against the manufacturer
of a certain make and model for a particular defect. If 5 percent or less of the cars of this make and model
have the defect, the firm will not recover its expenses. Therefore, the firm will handle the lawsuit only if it is
convinced that more than 5 percent of cars of this make and model have the defect. The firm plans to take a
random sample of 1,000 people who bought this car and ask them if they experienced this defect in their cars.
a) Define the parameter of interest and state the null and alternative hypotheses that the law firm should test.
b) In the context of this situation, describe Type I and Type II errors and describe the consequences of each of
these for the law firm.
A university is worried that it might not have sufficient housing for its students for the next academic year. It's
very expensive to build additional housing, so it is operating under the assumption (hypothesis) that the housing
it has is sufficient, and it will spend the money to build additional housing only if it is convinced it is necessary
(that is, it rejects its hypothesis).
a) For the university's assumption, what is the risk involved in making a Type-I error?
b)
For the university's assumption, what is the risk involved in making a Type-II error?
c)
What choice would you make, based on the risks of each type of error?