Elementary Statistics

1
Elementary Statistics

• Statistical Decision Making

2
Review

• Population Sample Statistical inference
• null hypothesis H0
• alternative hypothesis H1
• statistically significant

3
How to make decision?

• Assume the null hypothesis is true
• Assess whether or not the observed result is
  extreme or very unlikely.
• ------ What is extreme?
• unlikely, rare --- reject the null hypothesis
• not unusual favor or accept null hypothesis
• Note we want chance of making an error to be
  small !

4
How to set up our decision rule?
5
Frequency Plot

• Bag A has a total value - 560, while Bag B
  has a total value 1,890.

6
Hypotheses

• Collect data --- Observation
• Definition The number n of observations in a
  sample is called the sample size.
• N1

7
How will you decide?

• Think about it
• What if the voucher you select is 60?
• Would this observation lead you to think the
  shown bag is Bag A or Bag B?
• Why?
• How would you answer these questions if the
  voucher you select is 10?

8
Decision Rule
9
Calculate the chance

• Chance if Chance if
• Face Value Bag A Bag B
• -1,000 1/20 0
• 10 7/20 1/20
• 20 6/20 1/20
• 30 2/20 2/20
• 40 2/20 2/20
• 50 1/20 6/20
• 60 1/20 7/20
• 1,000 0 1/20

10
What is rare (unlikely) scenario?
11
Decision Rule 1
12
Rejection Region
13
Chance of errors?
14
Chance of error ?
15
What can we do? --- Change rule
16
Chance of error with rule 2
17
Need better result
18
Summary from this example
19
(No Transcript)
20
How to make decision? Classic Approach

• State Null and Alternate Hypothesis
• Write your decision rule
• Calculate ? , ?-- check your decision rule
• Look at your observed value
• Apply your decision rule to your observed value
  and make a decision on which hypothesis you want
  to support

21
More on the Direction of Extreme
22
More Example--- One-sided Rejection Region to the
Left
23
Decision and chance of error
24
(No Transcript)
25
Lets do it 1.7 continue

• Decision Rule 3 Reject if your selected voucher
  is less than and equal to 3 .
• The rejection region for Decision Rule 3 is
  _______________.
• The values for a and ß corresponding to Decision
  Rule 3 are as follows
• a
• ß

26
Review Make decision Classical Approach

• State Null and Alternate Hypothesis
• Write your decision rule
• Calculate ? , ? -- check your decision rule
• Look at your observed value
• Apply your decision rule to your observed value
  and make a decision on which hypothesis you want
  to support

27
P-value approach

• A significance level is fixed.!
• Idea
• Suppose H0 is true.
• Observe the value
• Computer the chance of getting the observed value
  and more extreme values
• --- This chance is called P-value
• Make decision by comparing the significance level
  ? and p-value.

28
P-value

• Calculate P-value

What is the p-value of getting a 50
voucher? What is the p-value of getting a 30
voucher?
29
How unusual is the data?

• The smaller the p-value, the more unusual the
  observed data.
• !! The smaller the p-value, the stronger is the
  evidence provided by the data against the null
  hypothesis H0

30
Classic Approach and P-value Approach

• Classical Decision Rule with ? 0.10
• Reject H0 if the selected voucher is ? 50
• P-value approach with ? 0.10
• Reject H0 if the p-value of the observed date is
  less than ? - the data are statistically
  significant

31
(a) For which study do the results show the most
support for the null hypothesis? Explain. (b)
Suppose Study A concluded that the data supported
the alternative hypothesis that the true average
lifetime is less than 54 months, but in fact the
true average lifetime is greater than or equal to
54 months. In our statistical language, would
this be called a Type I error or a Type II
error? (c ) If the results of Study A are
statistically significant, which hypothesis is
supported?
32
P-value and significance level
33
Once you have made a decision, the decision is
either right or wrong.
34
Two-side rejection region

• Direction of extreme to the left and to the
  right,

35
P-value for a Two-sided Rejection Region

• Bag E and Bag F, each bag contains 30 vouchers.

36
(a) Suppose the observed voucher value is 2.
Find the corresponding p-value. For the following
significance levels, are the data statistically
significant?