Transcript Section 4

Lesson 11 - 4
Using Inference to
Make Decisions
Knowledge Objectives
• Define what is meant by a Type I error.
• Define what is meant by a Type II error.
• Define what is meant by the power of a test.
• Identify the relationship between the power of a test
and a Type II error.
• List four ways to increase the power of a test.
Construction Objectives
• Describe, given a real situation, what constitutes a
Type I error and what the consequences of such an
error would be.
• Describe, given a real situation, what constitutes a
Type II error and what the consequences of such an
error would be.
• Describe the relationship between significance level
and a Type I error.
• Explain why a large value for the power of a test is
desirable.
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.
• Level of Significance – probability of making a Type I error, α
Hypothesis Testing: Four Outcomes
Reality
Do Not Reject H0
H0 is True
Correct
Conclusion
H1 is True
Type II
Error
Reject H0
Type I
Error
Correct
Conclusion
Conclusion
H0: the defendant is innocent
H1: the defendant is guilty
decrease α  increase β
increase α  decrease β
Type I Error (α): convict an innocent person
Type II Error (β): let a guilty person go free
Note: a defendant is never declared innocent; just not guilty
Hypothesis Testing: Four Outcomes
• We reject the null hypothesis when the alternative
hypothesis is true (Correct Decision)
• We do not reject the null hypothesis when the null
hypothesis is true (Correct Decision)
• We reject the null hypothesis when the null
hypothesis is true (Incorrect Decision – Type I error)
• We do not reject the null hypothesis when the
alternative hypothesis is true
(Incorrect Decision – Type II error)
Example 1
You have created a new manufacturing method for
producing widgets, which you claim will reduce the
time necessary for assembling the parts. Currently it
takes 75 seconds to produce a widget. The retooling of
the plant for this change is very expensive and will
involve a lot of downtime.
Ho :
Ha:
TYPE I:
TYPE II:
Example 1
Ho : µ = 75 (no difference with the new method)
Ha: µ < 75 (time will be reduced)
TYPE I: Determine that the new process reduces time
when it actually does not. You end up spending lots of
money retooling when there will be no savings. The plant
is shut unnecessarily and production is lost.
TYPE II: Determine that the new process does not reduce
when it actually does lead to a reduction. You end up not
improving the situation, you don't save money, and you
don't reduce manufacturing time.
Example 2
A potato chip producer wants to test the hypothesis
H0: p = 0.08 proportion of potatoes with blemishes
Ha: p < 0.08
Let’s examine the two types of errors that the producer
could make and the consequences of each
Type I Error:
Description: producer concludes that the p < 8% when its actually greater
Consequence: producer accepts shipment with sub-standard potatoes;
consumers may choose not to come back to the product after a bad bag
Type II Error:
Description: producer concludes that the p > 8% when its actually less
Consequence: producer rejects shipment with acceptable potatoes;
possible damage to supplier relationship and to production schedule
Example 3
A city manager’s staff takes a random sample of 400
emergency call response times that yielded x-bar = 6.48
minutes with a standard deviation of 2 minutes. The
manager wants to know if the response time decreased
from last year’s mean of 6.7 min?
Parameter to be tested: mean response time in min
Test Type:
left-tailed test
H0: Mean response time,  = 6.7 minutes
Ha: Mean response time,  < 6.7 minutes
Example 3
H0: Mean response time,  = 6.7 minutes
Ha: Mean response time,  < 6.7 minutes
Give the description and consequences of the two error
types:
The manager concludes that the response
Type I: times have improved, when they really have
not. No additional funding for improvement;
possible additional lives lost.
Type II: The manager concludes that the response
times still need to be improved, when they have
improved already. Additional funds spent
unnecessarily and morale might be lowered.
Graphical View of Error Types
Area to the
left of critical
value under
the right most
curve is the
Type I error
Area to the
right of critical
value under
the left most
curve is the
Type II error
• As the critical value of x-bar moves right α increases and β decreases
• As the critical value of x-bar moves left α decreases and β increases
• Need to identify the differences in errors and their consequences in a
given problem
Finding P(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 3
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 3
• 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 3
• 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 P(Type I error)
Example 3
• 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 3
• 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 P(Type II error)
H1
Critical Value (the same as before)
Example 3
• 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 3
● 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 3
• 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
Power and Type II Error
• Probability of a Type II error is β
• Power of the test is 1 – β
• P-value describes what would happen supposing the
null hypothesis is true
• Power describes what would happen supposing that
a particular alternative is true
Increasing the Power of a Test
• Four Main Methods:
– Increase significance level, 
– Consider a particular alternative that is farther
away from 
– Increase the sample size, n, in the experiment
– Decrease the population (or sample) standard
deviation, σ
• Only increasing the sample size and the
significance level are under the control of the
researcher
Comparisons
• P-value, (compared to ) assumes that H0 is
true
• Power, (1 - ) assumes that some alternative
Ha is true
Summary and Homework
• Summary
– A Type I error occurs if we reject H0 when in fact
its true; P(Type I) = α
– A Type II error occurs if we fail to reject HO when
in fact its false; P(Type II) = β
– Power of a significance test measures its ability to
detect an alternative hypothesis and is = 1 - β
– Increasing the power of a test can be done by
increasing sample size and by using a higher α
• Homework
– pg 727 – 735; 11.50, 51, 59-61