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5.4 Inference as Decision
Definition. A consumer may accept or reject a package of commodities
on the basis of the quality of a sample of the package. This is called
acceptance sampling.
Type I and Type II Errors
Definition. If we reject H0 (accept Ha ) when in fact H0 is true, this
is a Type I error. If we accept H0 (reject Ha ) when in fact Ha is true,
this is a Type II error. See Figure 5.17 (and TM-94).
Error Probabilities
Example 5.19. The mean diameter of a type of bearing is supposed to
be 2.000 centimeters (cm). The bearing diameters vary normally with
standard deviation σ = .010 cm. When a lot of the bearings arrives,
the consumer takes an SRS of 5 bearings from the lot and measures
their diameters. The consumer rejects the bearings if the sample mean
diameter is significantly different from 2 at the 5% level. This is a test
of the hypotheses:
H0 : µ = 2
Ha : µ = 2.
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To carry out the test, the consumer computes the z statistic
z=
x−2
√
.01/ 5
and rejects H0 if z > 1.96 A Type I error is to reject H0 when in fact
µ = 2. What about Type II errors? Because there are many values of µ
in Ha , we will concentrate on one value. The producer and the consumer
agree that a lot of bearings with mean diameter 2.015 cm should be
rejected. So a particular Type II error is to accept H0 when in fact
µ = 2.015 Figure 5.18 (and TM-95) shows how the two probabilities of
error are obtained from the two sampling distributions of x, for µ = 2
and for µ = 2.015. When µ = 2, H0 is true and to reject H0 is a Type I
error. When µ = 2.015, Ha is true and to accept H0 is a Type II error.
Definition. 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 H0 when H0 is in fact true.
Example 5.20. Let’s calculate the probability of a Type II error for
the previous example.
Step 1. Write the rule for accepting H0 in terms of x. This occurs
when
x−2
√ ≤ 1.96.
.01/ 5
or solving for x when 1.9912 ≤ x ≤ 2.0088.
−1.96 ≤
Step 2. Find the probability of accepting H0 assuming that the alternative is true. Take µ = 2.015 and standardize to find the probability:
P ( Type II error ) = P (1.9912 ≤ x ≤ 2.0088)
2

1.9912 − 2.015 x − 2.015
√
√
≤
.01/ 5
.01/ 5
2.0088 − 2.015
√
≤
.01 5
= P (−5.32 ≤ Z ≤ −1.39) = .0823.
= P
Figure 5.19 (and TM-96) show the relevant regions.
Power
Definition. The probability that a fixed level α significance test will
reject H0 when a particular alternative value of the parameter is true
is called the power of the test against that alternative. The power of
a test against any alternative is 1 minus the probability of a Type II
error for the alternative.
Example. The power of the test performed in the previous example
is 1 − .0823 = .9177.
Different Views of Statistical Tests
Note. The way of thinking about statistical tests called testing hypotheses involves:
1. State H0 and Ha just as in a test of significance. In particular, we
are seeking evidence against H0 .
2. Think of the problem as a decision problem, so that the probabilities
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of Type I and Type II errors are relevant.
3. Type I errors are more serious. So choose an α (significance level)
and consider only tests with probability of Type I error no greater
than α.
4. Among the tests, select one that makes the probability of a Type II
error as small as possible (that is, power as large as possible). If
this probability is too large, you will have to take a larger sample
to reduce the chance of error.
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