Homework
Chp 11. Use equations on formula sheet.
Page 417 # 5, 6, 8. (Use CI to make a decision).
Page 441 # 47, 49, 50. (Use CI to make . . . .)
Chp 15, Read 633 bottom, 634 middle.
Page 636 # 50 (Do part c to answer part b.)
# 51. (Do part c to answer part b.)
# 53, 54 (Review)
# 47, 48. (Give answer for p-value
(see answer to 47 ibob)
in relation to the a-level.)
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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Hypothesis Testing
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
2
Lesson Objectives
 Understand the “types of
errors” in decision making.
 Know what the a-level means.
 Learn how to use “p-values” and
confidence intervals for decision
making.
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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Court case
Hypothesis: Defendant is innocent.
Alternative: Defendant is guilty.
Decisions:
Reject
Innocence
Do not Reject
Innocence
Based on the sample
data.
Declare
“Guilty”
Person
goes to
jail!
Declare
“Not
Guilty”
Person
goes
free!
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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Types of Errors in a court case
Type I: Sending an innocent person
to jail.
Type II: Letting a guilty person go free.
a = level of risk deemed reasonable
for the occurrence of a Type I error.
= the point of “reasonable doubt.”
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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Types of Errors, in general
Type I: Concluding that the hypothesized
parameter value is wrong,
but in reality it is correct.
Type II: Not concluding that the
hypothesized parameter value is
wrong, but in reality it is incorrect.
a = level of risk, chosen by the user,
for allowing a Type I error to occur.
b = risk for making Type II error.
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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Potato Chip Inspection by FDA

Net weight of potato chip bags
should be 16.00 oz.

An FDA inspector will take a
random sample of 36 bags.
If the net weight is too low, the chip
company will be fined substantially.

From the FDA perspective,
what would the Type I and Type II
errors be (in words)?
Potato Chips; types of errors
Type I: Penalizing the potato chip
company when in reality they
were NOT cheating the consumer.
Type II: Not detecting that the potato
chip company was cheating the
consumers, when in reality they
were.
Which is more serious, from the FDA’s perspective?
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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Selecting an a-level
Chose an a-level that considers
the consequences of the
Type I and Type II errors.
a and b are inversely related;
as one goes up,
the other goes down,
but NOT by equal amounts.
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
9
Statistical Inference Methods:
Reject the hypothesized value if:
1. it is outside the confidence interval.
2. the p-value is less than the
user specified a-level. (p-value < a-level)
3. the calculated test statistic value
is in the “critical region.”
Three methods; each should give the same result.
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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Decisions are based on
the data.
Wrong decisions are the result
of
chance,
not mistakes.
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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(1-a)100% Confidence Interval Method
Hypothesized mean: 40.0
Two tailed test;
“Is the mean something
other than 40.0?”
Desired
a-level:
Size of
CI to use:
a
1-a
.10
.05
.01
.90
.95
.99
a
1 - 2a
.10
.05
.01
.80
.90
.98
Result: Each tail has half of a.
One tail test;
“Is the mean something
greater than 40.0?”
or
“Is the mean something
less than 40.0?”
Result: Each tail has the
full a-level. Use only ONE
tail for making a decision.
p-Value Method
The probability of observing a
future statistic value that is as big or
more extreme, in the direction(s) of interest,
than the value we just observed,
assuming that the hypothesized value
is the correct parameter.
Calculate p-value using the
most appropriate distribution.
Decision rule: If p-value < a-level,
reject
the
hypothesized
value.
M34- Hypothesis Testing 13
 Department of ISM, University of Alabama, 1992-2003
p-Value:
Hypo. mean: 40.0,
Two tailed test;
p-value / 2
“Is the mean something
2.6
other than 40.0?”
-4.0
Upper tail test;
“Is the mean something
greater than 40.0?”
-4.0
Lower tail test;
“Is the mean something
less than 40.0?”
-4.0
 Department of ISM, University of Alabama, 1992-2003
-3.0
37.4
-2.0
-1.0
X = 42.6
p-value /2
2.6
40
0.0
42.6
1.0
2.0
3.0
X
4.0
p-value
-3.0
-2.0
-1.0
40
0.0
42.6
1.0
2.0
3.0
X
4.0
p-value
-3.0
-2.0
-1.0
40 42.6
X
M34- Hypothesis Testing 14
0.0
1.0
2.0
3.0
4.0
Problem 1, using p-Value
Hypothesized mean: 40.0. Pick a = .05
.5000
Adjust machine if it’s off
.4332
.0668
in either direction.
X ~ N(  ?,   8.0)
Also
more extreme
More extreme
n  16
than 3.0 units
X ~ N( X  ?,  X  2.0)than 3.0 units
Sample results:
40 43.0
X
X = 43.0
What distribution
-1.50 0 1.50
Z
s = 7.2
should be used?
p-value = .0668 •2
43.0 – 40.0
Z=
= 1.50
= .1336
2.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
(p-value =.1336) > (a =.05); do not M34reject
40.0.
Hypothesis
Testing 15
 Department of ISM, University of Alabama, 1992-2003
Problem 1 with Confidence Interval
Hypothesized mean: 40.0. Pick a = .05
Adjust machine if it’s off

X ± Za/2
in either direction.
n
X ~ N(  ?,   8.0)
43.0 ± 1.96  2.0
n  16
X ~ N( X  ?,  X  2.0)
43.0 ± 4.92
Sample results:
(38.08, 47.92)
X = 43.0
s = 7.2
What distribution
should be used?
40.0 falls inside the C.I.;  do not reject 40.0.
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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Problem 2, using p-Value
Hypothesized mean: 40.0. Pick a = .05
FDA will fine company
if the mean is lower.
From Excel,
X ~ N(  ?,   ?)
.0320
n  16
More extreme
than 3.6 units
X ~ N( X  ?,  X  ?)
below.
Sample results:
X = 36.4
X
36.4 40
What distribution
-2.00 0
t
s = 7.2 Fromshould
be
used?
the t-table . . . .
36.4 – 40.0
p-value = more than .025,
t=
= -2.00
less than .050..
7.2 / 4
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
(p-value =.0320) < (a-level = .05);  reject
40.0.
M34- Hypothesis
Testing
 Department of ISM, University of Alabama, 1992-2003
4.0
17
Problem 2 with Confidence Interval
Hypothesized mean: 40.0. Pick a = .05
FDA will fine company
s
if the mean is lower.
X ± t.05, 15
n
X ~ N(  ?,   ?)
7.2
n  16
36.4 ± 1.753 
X ~ N( X  ?,  X  ?)
16
Sample results:
36.4 ± 3.1554
X = 36.4
What distribution
s = 7.2
should be used?
(33.245, 39.555)
40.0 falls outside the C.I.;  reject 40.0.
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
18
Does a person have ESP?
Experiment:
Two people in different rooms.
 “A” is shown one of five cards, selected
randomly. “A” transmits his thoughts.
 “B” selects the card he thinks is being
sent to him, and records it
 The process is repeated 20 times;
the cards are shuffled each time.
X = a count of the number correct.
X ~ Bino(n=20, p =.20)

 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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X = a count of the number correct.
X ~ Bino(n=20, p =.20)
 = np = 4.0 (Cannot use Normal approx.)
Hypothesized value: p = .20, a-level = .05
Data: “B” got 9 out 20 correct.
Does “B” do better than guessing?
Use Binomial Dist.
0.25
P(X = x)
p-value = P(X >= 9)
= 1 – P(X <= 8)
Use
= 1 – .9900
Table A.2
= .0100
0.20
Bino(20, 0.20)
0.15
0.10
0.05
0.00
0
1 2 3
4 5 6
7 8
9 10 11 12 13 14 15 16 17 18 19 20
X
Reject .20; she does better!
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
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The end
 Department of ISM, University of Alabama, 1992-2003
M34- Hypothesis Testing
21