Making Decisions

1
Chapter 1

• Section 1.1-1.3
• Making Decisions

2
Special Note

• Do not try to write down everything that is on
  these slides! That would be like copying
  everything you read in the text. The slides are
  available to be downloaded from the web site and
  all of the definitions are lifted directly from
  the text book.
• Paying attention to the topic and write down the
  key ideas in your notes is a much better
  strategy!

3
What is Statistics?

• Most basic A way to summarize information.
• Real Purpose A method for making decisions based
  upon data.

4
What is a Decision?

• Different sociological groups have different
  decision making methods. Methods which are likely
  to converge on a decision within a finite time
  interval range from dictatorship to direct
  democracy to consensus decision making. However,
  depending on how the methods are implemented in
  practice, any of these may lead to either no
  decision being made or to inconsistent decisions
  being made. (http//en.wikipedia.org)

5
Statistical Decision Making

• The statistical decision making process is well
  defined. Together with the scientific method,
  statistics provides us with a collection of
  principles and procedures for obtaining and
  summarizing information in order to make
  decisions. (Interactive Statistics)

6
The Scientific Method

• Formulate a theory
• Collect data to test theory
• Analyze the results
• Interpret results and make a decision
• Re-evaluate theory (peer review)

7
What is a Theory?

• Write down your definition.
• Share it with the person next to you.

8
Fundamental Idea

• A theory is rejected if it can be shown
  statistically that the data observed would be
  very unlikely to occur if the theory were in fact
  true.
• A theory is accepted if it is not rejected by the
  data.

9
The Butlers Guinea Pig

• The Butlers guinea pig started to get real fat.
  They were concerned that they had been over
  feeding her or that she had perhaps grown a
  tumor.
• Out popped three baby guinea pigs. The pet shop
  owner had assured them that the other pig in the
  pen with her was a female.

10
(No Transcript)
11
The Decision to Make

• Competing Theories The other guinea pig was
  female vs. the other guinea pig was male
• Collect Data Three baby guinea pigs
• Analyze Results The probability of the bunkmate
  being female is very small.
• Interpret and Make Decision Dont call the
  tabloids.

12
Example of Hypotheses

• Theory A study suggests the taking Glucosamine
  and Chondroitin will reduce joint pain for the
  majority of users.
• To test this theory we need to form competing
  hypotheses about the statement.
• The Null Hypothesis is the status quo, or
  prevailing view.
• The Alternate Hypothesis is the opposite of the
  null, the research hypothesis.

13
State the Null and Alternate

• The null hypothesis is denoted H0 and the
  alternate is given by H1
• H0 Taking Glucosamine and Chondroitin will not
  reduce joint pain for the majority of users (more
  than 50 of users).
• H1 Taking Glucosamine and Chondroitin will
  reduce joint pain in the majority of users (more
  than 50 of users).
• (Well let Glucosamine and Chondroitin be
  abbreviated as G/C from here)

14

• Example Average Life Span
• Suppose that you work for a company that produces
  cooking pots with an average live span of seven
  years. To gain a competitive advantage, you
  suggest using a new material that claims to
  extend the life span of the pots. You want to
  test the hypothesis that the average life span of
  the cooking pots made with this new material
  increases.
• H0The average life span of the new cooking pots
  is seven years.
• H1 The average life span of the new cooking
  pots is greater than seven years.

15
Lets Do It!

• Lets go get more practice setting up the null
  and alternate hypotheses.
• Page 5, 6 Lets Do It 1.1, 1.2
• Page 54 1.3

16

• Lets Do It! 1.1 -- Fair Die?
• In a famous die experiment, out of 315,672 rolls,
  a total of 106,656 resulted in a 5 or a 6.
  If the die is "fair, the true proportion of 5's
  or 6's should be 1/3.
• However, a close examination of a real die
  reveals that the "pips" are made by small
  indentations into the face of the die. Sides 5
  and 6 have more indentations than the other
  faces, and so these sides should be slightly
  lighter than the other faces, which suggests that
  the true proportions of 5's or 6's may be a bit
  higher than the "fair" value 1/3.
• State the appropriate null and alternative
  hypotheses for assessing if the data provide
  compelling evidence for the competing theory.
• H0 The die is fair, that is, the indentations
  have no effect, and the proportion of 5s or 6s
  is _____________.
• H1 The die is not fair, that is, the
  indentations have an effect, and the proportion
  of 5s or 6s is _____________.

17

• Lets Do It! 1.2 -- Stress can cause sneezes
• The article Stress can cause sneezes (The New
  York Times, January 21, 1997) suggest that stress
  doubles a persons risk of getting a cold. Acute
  stress, lasting maybe only a few minutes, can
  lead to colds. One mystery that is still
  prevalent in cold research is that while many
  individuals are infected with the cold virus,
  very few actually get the cold. On average, up to
  90 percent of people exposed to a cold virus
  become infected, meaning the virus multiplies in
  the body, but only 40 percent actually become
  sick. One researcher thinks that the accumulation
  of stress tips the infected person over into
  illness.
• The percentage of people exposed to a cold virus
  who actually get a cold is 40. The researcher
  would like to assess if stress increases this
  percentage. So, the population of interest is
  people who are under (acute) stress. State the
  appropriate hypotheses for assessing the
  researchers theory.

18
Recall Fundamental Idea

• A theory is rejected if it can be shown
  statistically that the data observed would be
  very unlikely to occur if the theory were in fact
  true. A theory is accepted if it is not rejected
  by the data.

19
Fundamental Definition

• Statistical Significance The data collected are
  said to be statistically significant if they are
  very unlikely to be observed under the assumption
  that the H0 is true. If data are statistically
  significant then our decision will be to reject
  the null hypothesis (H0)

20

• Lets Do It! 1.3 -- Complaints about Chips
• Last month, a large supermarket chain received
  many customer complaints about the quantity of
  chips in 16-ounce bags of a particular brand of
  potato chips. Wanting to assure its customers
  they were getting their money's worth, the chain
  decided to test the following hypotheses
  concerning the true average weight (in ounces) of
  a bag of such potato chips in the next shipment
  received from their supplier
• H0 Average weight is at least 16 ounces
• H1 Average weight is less than 16 ounces
• If there is evidence in favor of the alternative
  hypothesis, the shipment would be refused and a
  complaint registered with the supplier.
• Some bags of chips were selected from the next
  shipment and the weight of each selected bag was
  measured. The researcher for the supermarket
  chain stated that the data were statistically
  significant.
• What hypothesis was rejected?
• Was a complaint registered with the supplier?
• Could there have been a mistake? If so, describe
  it.

21
Recall Our G/C Hypotheses. Based on the Given
Data, Make a Decision

• If the proportion of subjects that report less
  joint pain is the same as with a placebo?
• If 75 of the subjects taking G/C report
  significantly less joint pain and only 35
  reported less pain that were taking the placebo?
• If the difference between G/C and the placebo was
  2?
• How large of a difference in proportion is needed
  for you to feel confident in rejecting the null
  hypothesis?

22
Couldve We Been Mistaken?

• Is it possible that if we concluded from our data
  that G/C worked that we could be wrong?
• Is it possible that if we concluded from our data
  that G/C didnt work that we could be wrong?

23
Types of Errors

• If we reject H0 when it was true weve made a
  Type I error
• If we fail to reject H0 when it is false, then
  weve made a Type II error
• For example, H0 Person is innocent H1 Person
  is guiltyExplain what a type I and type II error
  would be in this case.

24
The Truth
Your Decision Based Upon the Data
Alternate True
Null True
Type II Error
No Error
Null Accepted
No Error
Type I Error
Alternate Accepted
25
LDI 1.4 -- Which Error is Worse?

• H0 The water is contaminated.H1 The water is
  not contaminated.
• H0 The parachute works.H1 The parachute does
  not work.
• H0 A hostile country has weapons of mass
  destruction.H1 A hostile country does not have
  weapons of mass destruction.
• H0 The infant pain reliever has the stated
  amount of acetaminophen.H1 The infant pain
  reliever has more than the stated amount of
  acetaminophen.

26

• Lets Do It! 1.5 -- Testing a New Drug
• Two drugs are compared to see if the new one is
  more effective than the standard treatment.
• H0 The new drug is as effective as the standard
  drug.
• H1 The new drug is more effective than the
  standard drug.
• What are the two types of errors that you could
  make when deciding between these two hypotheses?
• Type I error
• Type II error
• What are the consequences of a Type I error?
• What are the consequences of a Type II error?
• Which error might be considered more severe from
  an ethical point of view?
• To know the true proportion of patients suffering
  from the disease that would be cured using the
  new drug, we would need to administer the new
  drug to all such patients. However, this is not
  possible. Why not?

27
Significance Level

• The probability of making a Type I error is
  called the level of significance. It is denoted
  by the Greek letter ? alpha
• The probability of making a type II error is
  denoted by the Greek letter ? beta

28
Definitions Please

• Population The entire group of objects or
  individuals under study
• Sample A part of the population that is actually
  used to get information
• Statistical Inference The process of making
  decisions about a population based upon the
  sample from that population

29
Homework 1

• Lets Do It (LDI) 1.11.5
• Exercises (EX) Page 54 1.1, 1.2, 1.3, 1.4, 1.5,
  1.6, 1.7, 1.9
• Read chapter 1, pages 139, 4954
• Spend some quality time with the Chapter Summary

30
Example

• Theory There are 4 blue balls and 1 yellow ball
  in the bag.
• Collect Data Pull a ball from bag, note color
  and replace it.
• Analyze the Results How many blue? How many
  yellow?
• Interpret and make Decision

31

• Section 1.4
• Its Not My Bag Baby

32
Whats in the Bag?
There are two bags -- call them Bag A and Bag B.
Each bag contains 20 vouchers of the same size
and shape. The contents of each bag, in terms of
the face value and the frequency of voucher
values, is described below
33
A Graphical Look

• We create a Frequency Plot for Bag A and Bag B.
  The x-axis is the data axis, the y-axis is the
  frequency.

34
The Problem

• We will be shown only one of the bags and be
  allowed to gather one piece of data from it. We
  then then have to decide whether to keep it or
  reject it. If we keep it and its Bag A we will
  need to pay 560. If it is Bag B we will win
  1890.

35
How are We Going to Decide?

• We are going to draw one voucher from the
  presented bag and use it to decide which bag, A
  or B.
• Our Hypotheses areH0 The shown bag is A (the
  bad one)H1 The shown bag is B

36
How to Decide

• We need to form a decision rule.
• Decision Rule A formal rule that states, based
  on the data obtained, when to reject the null
  hypothesis H0. Generally, it specifies a set of
  values based on the data to be collected, which
  are contradictory to the null hypothesis and
  which favor the alternate hypothesis.

37
More Basic

• Assume the null is correct
• Collect data
• If the data collected is very unlikely to occur
  in the distribution of the null but likely to
  occur in the distribution of the alternate, then
  reject the null hypothesis

38
Direction of Extreme

• The direction of extreme corresponds to the
  position of the values that are more likely under
  the alternate hypothesis than under the null
  hypothesis. If the larger values are more likely
  under H1 then the direction of extreme is to the
  right.

39
Decision Rule Take 1
Reject the null if the voucherselected is 60
or1000. That is,reject Ho if voucher gt 60
40
Decision Rule Definitions

• Rejection Region The set of values for which you
  would reject the null hypothesis Ho.
• Critical Value A value that marks the start of
  the rejection region.

41
What if Were Wrong?

• What is the chance of making a Type I error?
• What is the chance of making a Type II error?

42
Type I error can only occur if Ho is true.
Type Ierror
Type II error can only occur if H1 is true.
Type IIerror
43
(No Transcript)
44
Chances of Error

• So, our chance of type I error is 0.05, but our
  chances of type II error are 0.60. Are we willing
  to live with that large a chance of type II
  error, that is supporting the null when the
  alternate is true?
• Lets look at another version of the decision rule

45
Decision Rule 2

• Reject the null if the selected voucher is 50 or
  more otherwise accept the null that it is Bag A

46
(No Transcript)
47
Decision Rule 2

• Notice that we now have a chance of type I error
  of 0.10 and a chance of type II error of 0.30.
• Are we willing to live with this?
• Lets look at one more version

48
Decision Rule 3

• Reject the null if the selected voucher is 40 or
  more otherwise accept the null that it is Bag A

49
(No Transcript)
50
Decision Rule 3

• Notice that we now have a chance of type I error
  of 0.20 and a chance of type II error of 0.20.
• BIG DEAL Notice as we increase a the value of b
  decreases. The chances of making a type I or type
  II error are connected to each other. The other
  control is the sample size.

51
Summary

• Decision Rule gets you Significance level of
  aSelect a critical value (say 60) and a
  direction of extreme (gt 60) and you will get an
  a of 0.05
• Significance level of a gets you Decision
  Rule.Select a 0.10 then decision rule becomes
  Reject Ho if voucher is 50 or more.

52
More on the Direction of Extreme

• In our current example we have a one-sided
  rejection region, to the right. This is not the
  only possibility. We can have a rejection region
  to the left or we can have rejection regions on
  both the right and the left.

53
Example
BAG C
BAG D
54
GIVEN HYPOTHESES

• HO The shown bag is Bag C
• H1 The shown bag is Bag D
• Which is the direction of the most extreme value?
  Recall we are looking for the least likely value
  from the Null Hypothesis.

55
What is the Decision Rule?

• Reject the null hypothesis if the selected
  voucher is lt 1 otherwise accept the alternate
  hypothesis.

56
What is a and b ?

• a chance of rejecting Ho when it is true.
  This is the chance of selecting a 1 voucher from
  Bag C which is 1/15 or 0.067.
• b chance of accepting H0 when it is false.
  This is the chance of selecting a 2, 3, 4, or
  5 voucher from Bag D which is 10/15 or 0.667

57
Lets Do It!

• Page 23 LDI 1.6
• Example 1.6, Page 25 graphs.

58
Definition

• A rejection region is called one-sided if its set
  of extreme values are all in one direction, left
  or right
• A rejection region is called two-sided if its set
  of extreme values are in two directions, both
  left and right.

59
Question?

• What is the chance of seeing a 50 or more
  voucher selected given that we assume the bag is
  Bag A, that is we assume the null hypothesis is
  correct?
• This is the idea behind the p-value

60
The p-value

• The p-value is the chance, computed under the
  assumption that Ho is true, of getting the
  observed value plus the chance of getting all of
  the more extreme values.
• The p-value measures how likely the observed
  result is, or something even more extreme given
  the null is true.

61
Interpreting the p-value

• Small values of the p-value indicate that we have
  evidence against the null hypothesis. They
  indicate that the value drawn is in the rejection
  region.

62
Relationship Between p-value and the Significance
Level a (Big Deal)

• If p-val lt a then reject the null hypothesis,
  the data are statistically significant.
• If p-val gt a then accept the null hypothesis, the
  data are not statistically significant

63
Think About It and Lets Do It!

• Page 30.
• Page 32 LDI 1.9

64
Lets Do It

• Page 39 LDI 1.11

65
Chapter 1

• Section 1.6
• Is it Ethical?
• Is it Important?

66
What Does Significant Mean?

• Suppose a new medication to treat the crud has
  been developed and it is hypothesized it will
  cure it faster. What would the null and alternate
  hypothesis be?
• H0 The time to cure is the same for old and new
  treatments.H1 The new treatment cures faster.

67
You Do the Study

• You get a statistically significant difference in
  time to cure of 1/2 a day.
• The pharmaceutical company markets it as new and
  clinically shown to cure you faster.
• What do you think?

68
Relation Between Sample Size and Significance

• Case 1 With a large enough sample, even a small
  difference can be found to be statistically
  significant--that is, hard to explain by chance
  alone. This does not necessarily make it
  important.
• Think About It pages 50

69
Relation Between Sample Size and Significance

• Case 2 On the other hand, an important
  difference may not be statistically significant
  if the sample size is too small.
• Think About It pages 51

70
Relation Between Sample Size and Significance

• Example 1.13, page 52

71
Homework 2

• LDI 1.7, 1.8, 1.9, 1.10, 1.11
• Exercises Page 54 1.14, 1.16, 1.19, 1.21, 1.28,
  1.39, 1.46
• Read Section 2.1-2.5

72
Chapter Summary

• The goal of this chapter is to get you acquainted
  with the line of reasoning used in statistical
  decision making. What is the outline of this
  process?