Transcript Document

Lesson 10 - 7
Probability of a Type-II Error and
the Power of the Test
Objectives
• Determine the probability of making
a Type II error
• Compute the power of the test
Vocabulary
• Power of the test – value of 1 – β
• Power curve – a graph that shows the power of the test
against values of the population mean that make the null
hypothesis false.
Probability of Type II Error
• Determine the sample mean that separates the rejection region
from the non-rejection region
x-bar = μ0 ± zα · σ/√n
• Draw a normal curve whose mean is a particular value from the
alternative hypothesis, with the sample mean(s) found in step 1
labeled.
• The area described below represents β, the probability of not
rejecting the null hypothesis when the alternative hypothesis is
true.
a. Left-tailed Test: Find the area under the normal curve drawn
in step 2 to the right of x-bar
b. Two-tailed Test: Find the area under the normal curve
drawn in step 2 between xl and xu
c. Right-tailed Test: Find the area under the normal curve
drawn in step 2 to the left of x-bar
Example 1
The current wood preservative (CUR) preserves the
wood for 6.40 years under certain conditions. We have
a new preservative (NEW) that we believe is better, that
it will in fact work for 7.40 years
Ho : μ = 6.40 (our preservative is same as the current)
Ha: μ = 7.40 (new is significantly better than the current)
TYPE I:
TYPE II:
Example 1
• Our hypotheses
 H0: μ = 6.40 (our preservative is the same as the
current one)
 H1: μ = 7.40 (our preservative is significantly
better than the current one)
H0
H1
Example 1
• Type I error
– Assumes that H0 is true (that NEW is no better)
– Our experiment leads us to reject H0
H1
H0
Critical Value
This area is the Type I error
Example 1
• Type I errors
– Assumes that H0 is true (that NEW is no better)
– Our experiment leads us to reject H0
– Result – we conclude that NEW is significantly
better, when it actually isn’t
– This will lead to unrealistic expectations from our
customers that our product actually works better
Example 1
• Type II errors
– Assumes that H1 is true (that NEW is better)
– Our experiment leads us to not reject H0
H0
This area is the Type II error
H1
Critical Value (the same as before)
Example 1
• Type II errors
– Assumes that H1 is true (that NEW is better)
– Our experiment leads us to not reject H0
– Result – we conclude that NEW is not significantly
better, when it actually is
– This will lead to customers not getting a better
treatment because we didn’t realize that it was
better
Example 1
● Test Details
 We test our product on n = 60 wood planks
 We have a known standard deviation σ = 3.2
 We use a significance level of α = 0.05
● The standard error of the mean is
3.2
----- = 0.41
60
● The critical value (for a right-tailed test) is
6.40 + 1.645  0.41 = 7.08
Example 1
• The Type II error, β, is the probability of not rejecting
H0 when H1 is true
– H1 is that the true mean is 7.40
– The area where H0 is not rejected is where the sample mean
is 7.08 or less
• The probability that the sample mean is 7.08 or less,
given that it’s mean is 7.40, is
7.08 – 7.40
β = P( z < ------------------) = P(z < -0.77) = 0.22
0.41
• Thus β, the Type II error, is 0.22
• Power of the test is 1 – β = 0.78
Summary and Homework
• Summary
– The Type II error, β, is the probability of not rejecting
the null hypothesis when the alternative hypothesis
was actually true
– Type II errors can be computed only when the
alternative hypothesis is also an equality
– The power of a test, 1 – β, measures how well the
test distinguishes between the two hypotheses
• Homework
– pg 565 – 567; 3, 4, 7, 9