Ch7 Inference concerning means II

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Ch7 Inference concerning means II

• Dr. Deshi Ye
• yedeshi_at_zju.edu.cn

2
Review

• Point estimation calculate the estimated
  standard error to accompany the point
  estimate of a population.
• Interval estimation
• whatever the population, when the sample size
  is large, calculate the 100(1-a) confidence
  interval for the mean
• When the population is normal, calculate the
  100(1-a) confidence interval for the mean
• Where is the obtained from
  t-distribution with n-1 degrees of freedom.

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Review con.

• Test of Hypothesis
• 5 steps totally. Formulate the assertion that
  the experiment seeks to confirm as the
  alternative hypothesis
• P-value calculation

the smallest fixed level at which the null
hypothesis can be rejected.
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Outline

• The relation between tests and confidence
  intervals
• Operating characteristic curves
• Inference concerning two means
• Design Issues Randomization and Pairing

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7.6 The relation between tests and confidence
intervals

• Tests for two-sided alternatives and confidence
  intervals.
• Consider the 100(1-a) confidence interval for
  the mean given on p233

A level test of null hypothesis
Versus
Test critical region
6
Relations

• The accept region

Can also be expressed as
Relation confidence interval gives the interval
of plausible values for
So, if is contained in this interval, then
it cannot be rejected.
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Calculating Type II Error Probabilities
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Power of Test

• 1. Probability of Rejecting False H0
• Correct Decision
• 2. Designated 1 - ?
• 3. Used in Determining Test Adequacy
• 4. Affected by
• True Value of Population Parameter
• Significance Level ?
• Standard Deviation Sample Size n

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Finding Power Step 1
Reject
?
HypothesisH0 ?0 ? 368H1 ?0 lt 368
Do Not
Draw
Reject
? .05
?
368
?
X
0
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Finding Power Steps 2 3
Reject
?
???n 15/?25
HypothesisH0 ?0 ? 368H1 ?0 lt 368
Do Not
Draw
Reject
? .05
?
368
?
X
0
?
True Situation ?1 360
Draw
?
?
1-?
Specify
?
X
?
360
1
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Finding Power Step 4
Reject
?
???n 15/?25
HypothesisH0 ?0 ? 368H1 ?0 lt 368
Do Not
Draw
Reject
? .05
?
368
?
X
0
?
True Situation ?1 360
Draw
?
?
Specify
?
X
?
360
363.065
1
12
Finding PowerStep 5
Reject
?
???n 15/?25
HypothesisH0 ?0 ? 368H1 ?0 lt 368
Do Not
Draw
Reject
? .05
?
368
?
X
0
?
True Situation ?1 360
Draw
? .154
?
?
?
1-? .846
Specify
Z Table
?
X
?
360
363.065
1
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Power Curves
H0 ? ???0
H0 ? ???0
Power
Power
Possible True Values for ?1
Possible True Values for ?1
H0 ? ??0
Power
?? 368 in Example
Possible True Values for ?1
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7.7 Operating Characteristic Curves

• How about Type II errors?
• Review the example in Section 7.4 (P238)
• We investigate the probability of not rejecting
  (accepting) the null hypothesis under a range of
  values for

Probability of accepting the null hypothesis when
prevails
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If equals a value of the null hypothesis
is true, then is the probability of the
Type I error.
When has a value where the alternative
hypothesis is true, then is the
probability of a Type II error.
Example P238
If the prevailing population mean is
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Calculation

• Students are asked to calculate the table in P253

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OC curve

• The graph of for various value of
  
• is called operating characteristic curve, or
  simply OC curve.

Null Hypothesis
Alternative Hypothesis
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Type II error

• Table 8 at end of the textbook, the probabilities
  of Type II errors can be determined directly
  corresponding to the value

Quantity needed for use of Table 8
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Thinking Challenge
How Would You Try to Answer These Questions?

• Who Gets Higher Grades Males or Females?

• Which Programs Are Faster to Learn Windows or
  DOS?

D O S
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7.8 Inference concerning two means

• In many statistical problems, we are faced with
  decision about the relative size of the means of
  two or more populations.
• Tests concerning the difference between two means
  
• Consider two populations having the mean
• and and the variances of and
• and we want to test null hypothesis

Random samples of size
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Two Population Tests
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Testing Two Means

• Independent Sampling Paired Difference
  Experiments

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Two Population Tests
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Independent Related Populations
Independent
Related
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Independent Related Populations
Independent
Related

• 1. Different Data Sources
• Unrelated
• Independent

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Independent Related Populations
Independent
Related

• 1. Different Data Sources
• Unrelated
• Independent

• 1. Same Data Source
• Paired or Matched
• Repeated Measures(Before/After)

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Independent Related Populations
Independent
Related

• 1. Different Data Sources
• Unrelated
• Independent
• 2. Use Difference Between the 2 Sample Means
• ?X1 -?X2

• 1. Same Data Source
• Paired or Matched
• Repeated Measures(Before/After)

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Independent Related Populations
Independent
Related

• 1. Different Data Sources
• Unrelated
• Independent
• 2. Use Difference Between the 2 Sample Means
• ?X1 -?X2

• 1. Same Data Source
• Paired or Matched
• Repeated Measures(Before/After)
• 2. Use Difference Between Each Pair of
  Observations
• Di X1i - X2i

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Two Independent Populations Examples

• 1. An economist wishes to determine whether there
  is a difference in mean family income for
  households in 2 socioeconomic groups.
• 2. An admissions officer of a small liberal arts
  college wants to compare the mean SAT scores of
  applicants educated in rural high schools in
  urban high schools.

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Two Related Populations Examples

• 1. Nike wants to see if there is a difference in
  durability of 2 sole materials. One type is
  placed on one shoe, the other type on the other
  shoe of the same pair.
• 2. An analyst for Educational Testing Service
  wants to compare the mean GMAT scores of students
  before after taking a GMAT review course.

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Thinking Challenge
Are They Independent or Paired?

• 1. Miles per gallon ratings of cars before
  after mounting radial tires
• 2. The life expectancy of light bulbs made in 2
  different factories
• 3. Difference in hardness between 2 metals one
  contains an alloy, one doesnt
• 4. Tread life of two different motorcycle tires
  one on the front, the other on the back

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Testing 2 Independent Means
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Two Population Tests
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Test

• The test will depend on the difference between
  the sample means and if both samples
  come from normal population with known variances,
  it can be based on the statistic

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Theorem

• If the distribution of two independent random
  variables have the mean and
• and the variance and , then the
  distribution of their sum (or difference) has the
  mean (or ) and the
  variance

Two different sample of size
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Statistic for test concerning different between
two means
Is a random variable having the standard normal
distribution.
Or large samples
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Criterion Region for testing
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EX.

• To test the claim that the resistance of electric
  wire can be reduced by more than 0.05 ohm by
  alloying, 32 values obtained for standard wire
  yielded ohm and
  ohm , and 32 values obtained for alloyed
  wire yielded
• ohm and ohm
• Question At the 0.05 level of significance,
  does this support the claim?

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Solution

• 1. Null hypothesis

Alternative hypothesis
2. Level of significance 0.05
3. Criterion Reject the null hypothesis if Z gt
1.645
4. Calculation
5. The null hypothesis must be rejected.
6. P-value 1-0.9960.04 lt level of significance
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Critical values
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Type II errors

• To judge the strength of support for the null
  hypothesis when it is not rejected.
• Check it from Table 8 at the end of the textbook

The size of two examples are not equal
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Small sample size

• 2-sample t test.

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Criterion Region for testing (Statistic for
small sample )
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EX
The following random samples are measurements of
the heat-producing capacity of specimens of coal
from two mines Question use the 0.01 level of
significance to test where the difference between
the means of these two samples is significant.

• Mine 1 Mine 2
• 8260 7950
• 8130 7890
• 8350 7900
• 8070 8140
• 8340 7920
• 7840

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Solution

• 1. Null hypothesis

Alternative hypothesis
2. Level of significance 0.01

• Criterion Reject the null hypothesis if t gt 3.25
  or tlt -3.25

4. Calculation
5. The null hypothesis must be rejected.
6. P-value 0.004 lt level of significance 0.01
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Calculate it in Minitab
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Output

• Two-sample T for Mine 1 vs Mine 2
• SE
• N Mean StDev Mean
• Mine 1 5 8230 125 56
• Mine 2 6 7940 104 43
• Difference mu (Mine 1) - mu (Mine 2)
• Estimate for difference 290.000
• 99 CI for difference (43.293, 536.707)
• T-Test of difference 0 (vs not ) T-Value
  4.11 P-Value 0.004 DF 7

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• SE mean (standard error of mean) is calculated
  by dividing the standard deviation by the square
  root of n.
• StDev standard deviation .

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Confidence interval

• 100(1-a) confidence interval for

Where is based on
degrees of freedom.
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CI for large sample
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Matched pairs comparisons

• Question Are the samples independent in the
  application of the two sample t test?
• For instance, the test cannot be used when we
  deal with before and after data, where the data
  are naturally paired.
• EX A manufacturer is concerned about the loss of
  weight of ceramic parts during a baking step. Let
  the pair of random variables denote the
  weight before and weight after baking for the
  i-th specimen.

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Statistical analysis

• Considering the difference
• This collection of differences is treated as
  random sample of size n from a population having
  mean

indicates the means of the two responses are
the same
Null hypothesis
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EX

• The following are the average weekly losses of
  worker-hours due to accidents in 10-industrial
  plants before and after a certain safety program
  was put into operation
• Before 45 73 46 124 33 57 83 34 26 17
• After 36 60 44 119 35 51 77 29 24 11
• Question Use the 0.05 level of significance to
  test whether the safety program is effective.

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Solution

• 1. Null hypothesis

Alternative hypothesis
2. Level of significance 0.05
3. Criterion Reject the null hypothesis if t gt
1.833
4. Calculation
5. The null hypothesis must be rejected at level
0.05.
6. P-value 1-0.99850.0015 lt level of
significance
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Confidence interval

• A 90 confidence interval for the mean of a
  paired difference.
• Solution since n10 difference have the mean 5.2
  and standard variance 4.08,

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7.9 Design issues Randomization and Pairing
Randomization of treatments prevents
uncontrolled sources of variation from exerting a
systematic influence on the response
Pairing according to some variable(s) thought to
influence the response will remove the effect of
that variable from analysis
Randomizing the assignment of treatments within a
pair helps prevent any other uncontrolled
variables from influencing the responses in a
systematic manner.
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Data Collection and Analysis

• Practical project.
• 1. Data
• Your goal is to see how the American and National
  leagues compare on these variables.