Chapters 20-21 Summative Assessment Part 1

Question

A police officer uses a breathalyzer to test the blood alcohol content of drivers suspected of a DUI. The driver will be arrested if the officer believes that the driver is drunk, according to the results of the breathalyzer. The hypotheses can be stated as follows:

H₀ : The driver is not drunk.

H_a : The driver is drunk.

Which of the following best describes the power of the test?

a	The probability of arresting a driver who is drunk.
b	The probability of not arresting a driver who is not drunk.
c	The probability of not arresting a driver who is drunk.
d	The probability of arresting a driver who is not drunk.
e	The probability that the breathalyzer is working correctly.

A one-sided hypothesis test is to be performed with a significance level of 0.05. Suppose that the null hypothesis is false. If a significance level of 0.01 were to be used instead of a significance level of 0.05, which of the following would be true?

a

Neither the probability of a Type II error nor the power of the test would change.

b

The probability of a Type II error would decrease and the power of the test would increase.

c

Both the probability of a Type II error and the power of the test would increase.

d

The probability of a Type II error would increase and the power of the test would decrease.

e

Both the probability of a Type II error and the power of the test would decrease.

Answer 1

First they're asking for the power of the test. This is defined as the probability that a hypothesis test will correctly reject the null hypothesis. We can define this as 1-(P(TypeII error)), where a Type II error happens when we accept the null hypothesis when the test hypothesis is true. Let's go through the choices in the table.

a) Probability of arresting a driver who is drunk. In this case the Ha, that the driver is drunk, is true. And we are correctly arresting them.

^Power is about accepting H0, not Ha. So it's not this one.

b) Probability of not arresting a driver who is not drunk. So H0 is true, and we accept H0. This is the one!

(But let's look at the rest anyway)

c) Probability of not arresting a driver who is drunk. The Ha is true, but we've rejected it. This is a type II error.

d) Probability of arresting a driver who is not drunk. The H0 is true, but we've rejected it anyway. This is a Type I error.

e) Probability that the breathalyzer is working correctly. This is like a combination of power and confidence level (ca is 1-alpha, or 1-chance of making type I error)

Part 2:

What is significance level? The probability of a type I error (rejecting H0 when H0 is true). You'll also see it written as alpha. If we're changing alpha from 0.05 to 0.01, we're lowering the chance of a type I error and making it harder to reject H0.

If we make it harder to reject H0, though, we'll also have a higher chance of rejecting Ha when Ha is true. That is, we have a higher chance of Type II error. So now we can eliminate any answer choice that says P(Type II error) doesn't change or decreases.

This leaves us with c and d. What is the difference between them? Both c and d says P(Type II) error will increase. but they also say conflicting things about power.

Recall the definition of power. This is 1 - P(Type II error) so if P(Type II error) increases, power must decrease.

The one correct answer remaining is d