More statistical decisionmaking

1
More statistical decision-making
2
More statistical decision-making.

• Mid-semester course evaluation results.
• Hypothesis testing with coins (review).
• Hypothesis testing with coins what to do when
  you dont know the alternative hypothesis.
• Hypothesis testing with real data the sign test
  (Chapter 10).

Reminders Exam 2 is March 27!
3
Mid-semester course evaluations.
Teaching. Your instructor Mean
(SD) Communicates instructional matter
effectively. 3.7 (0.87) Establishes a class
environment that encourages student
participation. 4.3 (0.78) Conducts class
sessions in an organized, well-planned
manner. 4.3 (0.66) Communicates course
objectives and requirements explicitly. 4.1
(0.82) Has increased your appreciation,
understanding, and/or competence in this
subject. 3.4 (1.02) Has students apply
concepts to demonstrate understanding. 4.1
(0.59) Has genuine interest in students. 4.2
(0.72) Is enthusiastic about the subject of this
course. 4.4 (0.66) Motivates students to do
their best work. 4.1 (0.84) Challenges
students to think critically. 4.3
(0.6) Involves students in the learning
process. 4.2 (0.65) Establishes an
appropriate learning pace for the course. 3.2
(1.14) Assigns a reasonable workload for the
course. 3.8 (0.95) Is accessible to students
out of class, given class size. 3.8
(0.82) Maintains close agreement between stated
course objectives and what is actually
taught. 4.0 (0.91) Identifies readings/text
which enrich the quality of the course. 3.8
(0.89)  
4
Mid-semester course evaluations.
Evaluation. Your instructor Mean
(SD) Gives exams that reflect the material
covered in class. 3.6 (1.06) Uses appropriate
methods of evaluation to assess student
performance. 3.4 (0.99) Keeps students informed
of their progress. 3.7 (1.07) Fairly
evaluates of student learning. 3.5
(1.03) Grades student work promptly, considering
the class size. 4.5 (0.77) Provides helpful
comments and feedback where appropriate. 3.8
(1.08)   Overall. Please respond to the next
three items by circling the appropriate
number.   So far, how would you rate
the Course Instructor Learning you have
achieved so far 1. Poor 3.1 (0.77) 3.5
(0.90) 2.9 (0.95) 2. Adequate 3. Good 4. Very
good 5. Outstanding
5
Course changes and explanations.

• One week for Exam 2 (Turn in beginning of class
  on April 3).
• Why does the class cover so much so quickly?
  (Slow down a little by not covering chapter 14
  until after exam 2).
• Why dont you have a book of my slides?

6
Hypothesis testing with coins (review)

• Two friends, Otto and Alan.
• Each friend has 1 coin
• Ottos coin is fair -- p(Head) .50
• Alans coin is biased -- p(Head) .80
• While playing with their coins, the coins get
  mixed into a jar of your own fair coins, so now
  Alans coin is lost.
• How can you tell if a coin in the jar is fair or
  biased?

7
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• HOtto Coin is fair p(Head) 0.50
• HAlan Coin is biased p(Head) 0.80
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which your
  decision hinges X ??

Probability distribution of X given that HO is
true.
8
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• HOtto Coin is fair p(Head) 0.50
• HAlan Coin is biased p(Head) 0.80
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of calling a
  fair coin biased a lt .05.
• 6. Define value of statistic upon which your
  decision hinges X gt 15.
• 7. Perform experiment/collect data X 18
• 8. Compare observed statistic to critical value.
  Is 18 in rejection region?
• 9. Decide ??
• 10. Draw conclusion using at least one complete
  sentence

9
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• HOtto Coin is fair p(Head) 0.50
• HAlan Coin is biased p(Head) 0.80
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of calling a
  fair coin biased a lt .05.
• 6. Define value of statistic upon which your
  decision hinges X gt 15.
• 7. Perform experiment/collect data X 18
• 8. Compare observed statistic to critical value.
  Is 18 in rejection region?
• 9. Decide Reject HO
• 10. Draw conclusion using at least one complete
  sentence Based on the results of this
  experiment, the coin is a biased coin.

10
Two types of errors you might make.
Reality
Ottos fair coin
Alans biased coin
Error! Calling a fair coin a biased coin.
Correct! Calling a biased coin a biased coin.
Not Ottos fair coin
Type I error, or a
Decision
Error! Calling a biased coin a fair coin.
Correct! Calling a fair coin a fair coin.
Ottos fair coin
Type II error, or b
11
What errors might you make and with what
probability?
Probability distribution of X given that HO is
true.
a You decide that you have a biased coin when
you dont.
No Reject!
Reject!
b
You decide that you have a fair coin when you
dont.
Probability dist. of X given that HA is true.
12
Two types of errors you might make.
Reality
p(Head) 0.50
p(Head) 0.80
Power
Error! Calling a fair coin a biased coin.
Correct! Calling a biased coin a biased coin.
Coin is biased
p(Correct) .8042
Decision
p(Type I error) .0207
Error! Calling a biased coin a fair coin.
Correct! Calling a fair coin a fair coin.
Coin is fair
p(Correct) .9793
p(Type II error) .1958
Note everything you need to know about power is
here and on pages 227-231.
13
Hypothesis testing with coins when the
alternative hypothesis is unknown.

• Alternative hypothesis? Whats that?
• Two hypotheses (mutually exclusive and
  exhaustive)
• H0 Null hypothesis chance alone is operating
• HA Alternative hypothesis something besides
  chance is operating
• Ottos coin represents H0 because it obeys the
  rules of probability that you would expect of a
  coin that is a member of a population where
  p(Head) 0.50
• Alans coin represents HA (or H1) because it
  behaves like a coin that was selected from a
  whole different alternative population
  something has acted upon it so that p(Head)
  0.80.

14
Hypothesis testing with coins when the
alternative hypothesis is unknown.

• In the problems that we have done, HA was fixed
  p(Head) 0.80.
• How do you carry out a hypothesis test if you
  know that one coin is biased towards heads, but
  dont know how much?
• H0 p(Head) 0.50
• HA p(Head) gt 0.50
• How do you carry out a hypothesis test if you
  know that one coin is unfair, but dont know how
  much or in which direction?
• H0 p(Head) 0.50
• HA p(Head) 0.50

A directional hypothesis
A non-directional hypothesis
15
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses (Mutually exclusive and
  exhaustive)
• H0 Coin is fair p(Head) 0.50
• HA Coin is biased toward heads p(Head) gt
  0.50
• Write H0 and HA down on a piece of paper. Ask
  yourself, are they mutually exclusive and
  exhaustive?
• If the answer is no, and you are sure that you
  have HA correct, alter H0 to make the two
  hypotheses mutually exclusive and exhaustive.

16
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses (Mutually exclusive and
  exhaustive)
• H0 Coin is fair p(Head) lt 0.50
• HA Coin is biased toward heads p(Head) gt 0.50

17
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• H0 Coin is fair p(Head) lt 0.50
• HA Coin is biased p(Head) gt 0.50
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of calling a
  fair coin biased a lt .05.
• 6. Define value of statistic upon which your
  decision hinges

Probability distribution of X given that HO is
true.
18
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• H0 Coin is fair p(Head) lt 0.50
• HA Coin is biased p(Head) gt 0.50
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of calling a
  fair coin biased a lt .05.
• 6. Define value of statistic upon which your
  decision hinges X gt 15.
• 7. Perform experiment/collect data X 14
• 8. Compare observed statistic to critical value.
  Is 14 in rejection region?
• 9. Decide ??
• 10. Draw conclusion using at least one complete
  sentence

19
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• H0 Coin is fair p(Head) lt 0.50
• HA Coin is biased p(Head) gt 0.50
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of calling a
  fair coin biased a lt .05.
• 6. Define value of statistic upon which your
  decision hinges X gt 15.
• 7. Perform experiment/collect data X 14
• 8. Compare observed statistic to critical value.
  Is 14 in rejection region?
• 9. Decide Fail to reject HO
• 10. Draw conclusion using at least one complete
  sentence Based on the results of this
  experiment, the coin is not a biased coin.

20
What errors might you make and with what
probability?
Probability distribution of X given that the most
extreme value of HO is true.
a You decide that you have a biased coin when
you dont.
No Reject!
Reject!
Only 1 of an infinite number of possibilities!
Probability dist. of X given that HA is P(Head)
0.80.
21
What errors might you make and with what
probability?
Probability distribution of X given that the most
extreme value of HO is true.
a You decide that you have a biased coin when
you dont.
No Reject!
Reject!
Probability dist. of X given that HA is gt 0.50.
?
22
Two types of errors you might make.
Reality
p(Head) lt 0.50
p(Head) gt 0.50
Power
Error! Calling a fair coin a biased coin.
Correct! Calling a biased coin a biased coin.
Coin is biased
p(Correct) ????
Decision
p(Type I error) .0207
Error! Calling a biased coin a fair coin.
Correct! Calling a fair coin a fair coin.
Coin is fair
p(Correct) .9793
p(Type II error) ????
23
Hypothesis testing with coins when the
alternative hypothesis is unknown.

• How do you carry out a hypothesis test if you
  know that one coin is biased towards heads, but
  dont know how much?
• H0 p(Head) lt 0.50
• HA p(Head) gt 0.50
• How do you carry out a hypothesis test if you
  know that one coin is unfair, but dont know how
  much or in which direction?
• H0 p(Head) 0.50
• HA p(Head) 0.50

A directional hypothesis
A non-directional hypothesis
24
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses (Mutually exclusive and
  exhaustive)
• H0 Coin is fair p(Head) 0.50
• HA Coin is biased toward heads p(Head)
  0.50
• Write H0 and HA down on a piece of paper. Ask
  yourself, are they mutually exclusive and
  exhaustive?
• If the answer is no, and you are sure that you
  have HA correct, alter H0 to make the two
  hypotheses mutually exclusive and exhaustive.

25
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses (Mutually exclusive and
  exhaustive)
• H0 Coin is fair p(Head) 0.50
• HA Coin is biased toward heads p(Head) 0.50

26
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• H0 Coin is fair p(Head) 0.50
• HA Coin is biased p(Head) 0.50
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of calling a
  fair coin biased a lt .05.
• 6. Define value of statistic upon which your
  decision hinges

Probability distribution of X given that HO is
true.
27
What does the rejection region mean?

• The rejection region defines the events that are
  so unlikely to occur given that HO is true (p
  .50) that, if they do occur, you are willing to
  reject the idea that H0 is true in favor of the
  belief that HA is true.

• What values of X
• Would make you
• Reject the non-directional
• H0 that p(head) .50?
• High values of X?
• Low values of X

Probability distribution of X given that HO is
true.
28
What does the rejection region mean?

• The rejection region defines the events that are
  so unlikely to occur given that HO is true that,
  if they do occur, you are willing to reject the
  idea that HO is true in favor of the belief that
  HA is true.

Always must limit a lt .05!
Probability distribution of X given that HO is
true.
No Reject!
Reject!
Reject!
p(Xlt5)H0 .0207
p(Xgt15)H0 .0207
29
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• H0 Coin is fair p(Head) 0.50
• HA Coin is biased p(Head) 0.50
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of calling a
  fair coin biased a lt .05.
• 6. Define value of statistic upon which your
  decision hinges 5 gt X gt 15.
• 7. Perform experiment/collect data X 4
• 8. Compare observed statistic to critical value.
  Is 4 in rejection region?
• 9. Decide ??
• 10. Draw conclusion using at least one complete
  sentence

30
Decision-making steps

• 1. Define problem Is a given coin fair or biased
  (assume fair)?
• 2. Define hypotheses
• H0 Coin is fair p(Head) 0.50
• HA Coin is biased p(Head) 0.50
• 3. Define experiment Take a coin, flip it 20
  times.
• 4. Define statistic X, the number of Heads
  observed in 20 flips.
• 5. Define acceptable probability of calling a
  fair coin biased a lt .05.
• 6. Define value of statistic upon which your
  decision hinges 5 gt X gt 15..
• 7. Perform experiment/collect data X 4
• 8. Compare observed statistic to critical value.
  Is 4 in rejection region?
• 9. Decide Reject H0
• 10. Draw conclusion using at least one complete
  sentence Based on the results of this
  experiment, the coin is not a fair coin.

31
What errors might you make and with what
probability?
a You decide that you have a biased coin when
you dont. p .0207
Probability distribution of X given that HO is
true.
No Reject!
Reject!
Reject!
a You decide that you have a biased coin when
you dont. p .0207
Probability dist. of X given that HA is gt 0.50.
?
32
Two types of errors you might make.
Reality
p(Head) 0.50
p(Head) 0.50
Power
Error! Calling a fair coin a biased coin.
Correct! Calling a biased coin a biased coin.
Coin is biased
p(Correct) ????
Decision
p(Type I error) .0414
Error! Calling a biased coin a fair coin.
Correct! Calling a fair coin a fair coin.
Coin is fair
p(Correct) .9586
p(Type II error) ????
33
Applying hypothesis testing with coins to science.

• Coin are an easy way to introduce hypothesis
  testing.
• Also an example of a real and valuable
  statistical procedure the sign test
• The sign test is used when you have two sets of
  scores that are paired because they are scores
  collected from the same person while he/she is
  participating in two different conditions.
• Known as repeated measures, within-subjects
  or correlated measures designs.
• Goal of the sign test is to determine if scores
  collected under one condition differ from those
  collected under the other condition
• Significant as opposed to random
  (non-significant) difference.
• Do any differences between conditions reflect
  chance variation (H0) or do the differences
  suggest that, under one of the conditions, people
  were acting like they were drawn from another
  population (HA).

34
Example of sign test nicotine and heart rate.

• Nicotine is a psychomotor stimulant should
  increase heart rate.
• Nicotine is present in tobacco and tobacco smoke
  that smokers inhale.
• If tobacco smokers are taking in nicotine, their
  heart rate should increase during smoking as
  compared to before smoking.
• Heart rate fluctuates naturally over time (random
  variability).
• Sometimes the heart beats faster
• Sometimes the heart beats slower
• Tobacco smoking is expected to increase heart
  rate beyond this natural random variability.

35
One way to examine the effect of tobacco smoke on
HR

• Recruit 32 smokers.
• Ask them to abstain from smoking for 8 hours.
• Measure HR before and during smoking.
• Condition 1 Measure HR for 5 minutes before
  smoking (HRbefore)
• Condition 2 Measure HR for 5 minutes during
  smoking (HRduring)

36
Some questions to think about HA.

• If tobacco smoking does increase HR (systematic
  change)
• Which should be greater HRbefore or HRduring?
• If you subtracted HRbefore from HRduring
  (HRduring HRbefore) would you expect the
  resulting difference score to be positive () or
  negative (-)?
• Across all subjects would you expect more plusses
  or more minuses?
• Is p(Plus) greater than, less than, or equal to
  0.5?

37
Some questions to think about H0.

• If tobacco smoking does not increase HR (random
  variation)
• Which should be greater HRbefore or HRduring?
• If you subtracted HRbefore from HRduring
  (HRduring HRbefore) would you expect the
  resulting difference score to be positive () or
  negative (-)?
• Across all subjects would you expect more plusses
  or more minuses?
• Is p(Plus) greater than, less than, or equal to
  0.5?

38
Decision-making steps

• 1. Define problem Does tobacco smoking increase
  HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Tobacco smoking does not increase HR
  p(Plus) 0.50
• HA Tobacco smoking increases HR p(Plus)
  0.50
• 3. Define experiment 32 smokers, measure HR
  before and during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which your
  decision hinges P ??

39
(No Transcript)
40
Rejection region these outcomes make H0 very
difficult to believe p(P gt 22) 0.0249
41
Decision-making steps

• 1. Define problem Does tobacco smoking increase
  HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Tobacco smoking does not increase HR
  p(Plus) lt 0.50
• HA Tobacco smoking increases HR p(Plus) gt
  0.50
• 3. Define experiment 32 smokers, measure HR
  before and during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges Reject H0 if P gt 22
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence

42
Decision-making steps

• 1. Define problem Does tobacco smoking increase
  HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Tobacco smoking does not increase HR
  p(Plus) lt 0.50
• HA Tobacco smoking increases HR p(Plus) gt
  0.50
• 3. Define experiment 32 smokers, measure HR
  before and during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges Reject H0 if P gt 22
• 7. Perform experiment/collect data P 30
• 8. Compare observed statistic to critical value.
  Is 30 in rejection region?
• 9. Decide Reject H0
• 10. Draw conclusion using at least one complete
  sentence Based on these results, tobacco smoking
  does increase heart rate.

43
Example of sign test denicotinized cigs and HR.

• Normal tobacco smoke increases HR in abstinent
  smokers.
• Is it the nicotine or some other smoke
  constituent that causes this HR increase?
• What would happen if smokers smoke denicotinized
  tobacco cigarettes instead of normal tobacco
  cigarettes? Would heart rate increase, decrease,
  or stay the same?

44
One way to examine the effect of tobacco smoke on
HR

• Recruit 32 smokers.
• Ask them to abstain from smoking for 8 hours.
• Measure HR before and during smoking of
  denicotinized tobacco cigarettes.
• Condition 1 Measure HR for 5 minutes before
  smoking (HRbefore)
• Condition 2 Measure HR for 5 minutes during
  smoking (HRduring)
• Hypothesize some change in HR, but unknown
  direction (hint a non-directional hypothesis!).

45
Decision-making steps

• 1. Define problem Does smoking denicotinized
  tobacco influence HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Denicotinized tobacco does not influence
  HR p(Plus) 0.50
• HA Denicotinized tobacco influences HR
  p(Plus) 0.50
• 3. Define experiment 32 smokers, measure HR
  before and during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which your
  decision hinges P ??

46
(No Transcript)
47
Rejection region these outcomes make H0 very
difficult to believe p(P gt 22) 0.0249
Rejection region these outcomes make H0 very
difficult to believe p(P gt 22) 0.0249
48
Decision-making steps

• 1. Define problem Does smoking denicotinized
  tobacco influence HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Denicotinized tobacco does not influence
  HR p(Plus) 0.50
• HA Denicotinized tobacco does influence HR
  p(Plus) 0.50
• 3. Define experiment 32 smokers, measure HR
  before and during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges
• Reject H0 if 10 gt P gt 22
• 7. Perform experiment/collect data
• 8. Compare observed statistic to critical value.
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence

49
Decision-making steps

• 1. Define problem Does smoking denicotinized
  tobacco influence HR?
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0 Denicotinized tobacco does not influence
  HR p(Plus) 0.50
• HA Denicotinized tobacco does influence HR
  p(Plus) 0.50
• 3. Define experiment 32 smokers, measure HR
  before and during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges
• Reject H0 if 10 gt P gt 22
• 7. Perform experiment/collect data P 19
• 8. Compare observed statistic to critical value.
  Is 19 in rejection region?
• 9. Decide Fail to reject H0
• 10. Draw conclusion using at least one complete
  sentence Based on these results, there is no
  evidence to support the idea that smoking
  denicotinized tobacco influences heart rate.

50
Example of sign test Phillip Morris Accord and
HR.

• Normal tobacco smoke increases HR in abstinent
  smokers.
• Denicotinized tobacco smoke does not influence
  heart rate.
• Taken together, these results suggest that
  nicotine is the constituent in tobacco smoke that
  increases HR.
• Phillip Morris has developed a new product, the
  Accord, that heats but does not burn tobacco.
• Does the Accord influence HR, thus suggesting
  nicotine delivery?

51
One way to examine the effect of tobacco smoke on
HR

• Recruit 32 smokers.
• Ask them to abstain from smoking for 8 hours.
• Measure HR before and during smoking of Accord.
• Condition 1 Measure HR for 5 minutes before
  smoking (HRbefore)
• Condition 2 Measure HR for 5 minutes during
  smoking (HRduring)
• Hypothesize some change in HR, but unknown
  direction.

52
Accord study fill in the blanks!

• 1. Define problem
• 2. Define hypotheses with respect to sign of
  (HRduring HRbefore)
• H0
• HA
• 3. Define experiment 32 smokers, measure HR
  before and during smoking.
• 4. Define statistic P, the number of Plusses
  observed with 32 subjects.
• 5. Define acceptable probability of Type I error
  a lt .05.
• 6. Define value of statistic upon which decision
  hinges
• 7. Perform experiment/collect data P
• 8. Compare observed statistic to critical value.
  Is observed value of P in rejection region?
• 9. Decide
• 10. Draw conclusion using at least one complete
  sentence

53
Future notes.

• Chapter 12
• Logic of using the normal curve for hypothesis
  testing.
• Hypothesis testing with one sample when m and s
  are known z.
• Skip pages 283-289.
• Chapter 13
• Hypothesis testing with one sample when m is
  known but s is unknown t.
• Skip pages 304-312.
• Exam 2
• will cover Chapters 8, 10, 12, and 13 (skipping
  the pages noted).
• handed out on 3/27 and due back at the beginning
  of class on 4/3.
• Chapter 14 on April 3rd!