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Transcript
AP STATISTICS
LESSON 10 – 4
( DAY 1 )
INFERENCE AS DECISION
ESSENTIAL QUESTION:
What are the two types of
errors made when making
decisions using inference?
Objectives:
• To identify type I and type II errors.
• To calculate type I and type II errors.
Inference as Decision
Tests of significance assess the strength of
evidence against the null hypothesis.
The alternative hypothesis ( the statement we
seek evidence for) enters the test only to help
us see what outcomes count against the null
hypothesis.
Using significance tests with fixed α, however,
suggests another way of thinking.
A level of significance α chosen in advance
points to the outcome of the test as a decision.
The transformation from measuring the strength
of evidence to making decisions is not a small
step.
Acceptance Sampling
There are circumstances that call for a
decision or action as the end result of
inference. Acceptance sampling is one
such circumstance.
We will use acceptance sampling to show
how a different concept – inference as
decision – changes the reasoning used in
tests of significance.
Type I and Type II Errors
There are simply two hypotheses, and we must
accept one and reject the other.
It is convenient to continue to call the two
hypotheses Ho and Ha , but Ho no longer has the
special status (the statement we try to find
evidence against) that it had in tests of
significance.
In the acceptance sampling problem, we must
decide between Ho and Ha.
Type I and Type II Errors
(continued…)
• If we reject Ho ( accept Ha) when in fact Ho
is true, this is a Type I error.
• If we accept (reject Ha ) H0 when in fact Ha
is true, this is a Type II error.
The Two Types of Error in
Testing Hypotheses
Error Probabilities
We assess any rule for making decisions by
looking at the probabilities of the two types of
errors. This is in keeping with the idea that
statistical inference is based on asking, “ What
would happen if I used this procedure many
times?”
Significance tests with fixed level α give a rule for
making decisions, because the test either
rejects Ho or fails to reject it.
We then describe the performance of a test by
the probabilities of type I and type II errors.
Example 10.21
Page 595
Are These Potato Chips Too Salty?
Steps for finding type I and type II errors:
Step 1: To find the type I error you use the α or
significance level = type I error.
Finding type II error:
Step 2: find Z values for α level:
Ho : μ = 2
Ha : μ ≠ 2
Example 10.21
(continued…)
z < -1.96 or z > 1.96
Step1: write the rule in terms of x.
2 – 1.96σ/√ n ≤ x ≤ 2 + 1.96σ/√ n
1.9723 ≤ x ≤ 2.027
Step 2: find the probability of accepting Ho
assuming that the alternative is true. Take μ =
2.05 and standardize to find the probability.
Figure 10.20
Page 596
• The light shaded area is a Type 1 error (the
probability of rejecting Ho: μ = 2 when in fact μ = 2.
• The probability of a Type II error (dark shaded area)
is the probability of accepting Ho when in fact μ =
2.05.
Significance and
Type I Error
The significance level α of any fixed level
test is the probability of a Type I error.
That is, α is the probability that the test will
reject the null hypothesis Ho when Ho is in
fact true.
Interpreting Type II Error
The probability of a type
II error in example 10.22
is .0571. This tells us
that this test will lead
us to fail to reject Ho : μ
= 2 for about 6% of all
batches of chips with a
μ = 2.05. In other words,
we will accept 6% of
batches of potato chips
so bad that their mean
salt content is 2.05 mg.