Download Inference as Decision

AP Statistics
10.4 Errors, Power, and Decisions
Date ___________________
Inference as Decision
Tests of significance assess the evidence against the null hypothesis. We measure
evidence by the p-value, a probability found under the assumption that the null
hypothesis is true. Using a fixed level significance with our test can be considered as
making a decision based on the outcome.
We do our best to make correct decisions, but sometimes we will be wrong. There are
two types of errors we can make. Make a sketch to represent each type of error.
TYPE I ERROR: We reject the null hypothesis when it is in fact true. (We reject a true
null.)

Consider, for example,  = 0.05 and a two-tailed test. Make a sketch and state the
probability of committing a type I error.
TYPE II ERROR: We fail to reject the null hypothesis when in fact the alternative is
true. (We fail to reject a false null.)
Truth about a population
Decision
based
on a
sample
H o True
Reject H o
Fail to reject H o
(Accept H o )
H a True
Probability of Type I, Type II errors and the Power of a Test
Example 1: Consider a fixed level two-tailed test of significance with α = .05 and a
population of soda cans with weights μ = 12 and σ = 0.5 ounces. Let n = 25.
We know that P (Type I error) = , the level of significance. Calculating the probability
of a type II error, however, requires more of us. We will need to calculate the probability
of a Type II error when a particular alternative is true. Let’s consider that a particular
alternative, for example, μ = 12.2, is really true. We will be calculating the probability of
failing to reject the null hypothesis (μ = 12) when this alternative is true (μ = 12.2).

Make a sketch to represent this situation.
1. When do we fail to reject the null hypothesis? (For what values of z?)
2. For which sample values (x bar) do we fail to reject the null hypothesis?
3. What is the probability that x bar has the values found above if the true mean is 12.2?
This is the probability of a type II error.
POWER OF A TEST
The Power of a test is the probability that a fixed level α significance test will
reject the null hypothesis H o when a particular value of the parameter is true. (In other
words, the power of a test is the probability of rejecting a false null.)
POWER = 1 – P (type II error)

What is the power of the test in the example above? (What is the probability that
the test will reject the null hypothesis that μ= 12 when in fact μ = 12.2?)
There are several things we can do to increase the power of a test. It is a generally
accepted industry standard that power should, roughly, be greater than or equal to 0.80.
How can we increase the power of a test?
1.
2.
3.
4.
Example 2: Recalculate P (type II error) and the POWER if n = 100 in the same cola
filling problem.
 Make a sketch to represent this situation.
1. When do we fail to reject the null hypothesis? (For what values of z?)
2. For which sample values (x bar) do we fail to reject the null hypothesis?
3. What is the probability that x bar has the values found above if the true mean is 12.2?
This is the probability of a type II error. What is the power of the test now?
HW: p. 599 #69
You are thinking about opening a restaurant and are searching for a good location. From
research you have done, you know that the mean income of those living near the
restaurant must be over $45,000 to support the type of restaurant you wish to open. You
decide to take a simple random sample of 50 people living near one potential location.
Based on the mean income of this sample, you will decide whether or not to open a
restaurant there. A number of similar studies have shown that σ= $5,000.
a) Describe in context the two types of errors you could make. Identify which is Type I
and which is Type II.
b) Which of the errors is most serious? Explain the implication of each error.
c) State the null and alternate hypotheses.
d) If you had to choose one of the “standard” significance levels for your significance
test, would you choose α to be 0.01, 0.05, or 0.10? Justify your choice.
e) Using α = 0.05, how high will the sample mean have to be before you decide to open a
restaurant in that area?
f) If the mean income in a certain area is $47,000, how likely are you not to open a
restaurant in that area? What is the probability of a Type II error?
g) What is the power of your test? Is this acceptable?