Chapters 8 and 9 Statistical Inference
Types of statistical Inference:
 Point Estimation (Chapters 7, 8, 9)
 Interval Estimation (Chapters 7, 8, 9)
 Hypothesis Testing (Chapters 8, 9)
Point Estimation:
Population
Parameters
Unbiased Estimators
(Statistics)
1 n
X   X i = Sample Mean
 = Mean
n i 1
X
= Sample Proportion
p̂

p = Proportion
n
 = Std. Dev S = Sample Std. Dev. (Small Bias)
1 – 2
d = 1 – 2
p1 – p2
X1  X 2
1 n
X d   di
n i 1
1 n
  ( X 1i  X 2i )
n i 1
ˆp1  ˆp2
where ˆp1 
X1
X
, ˆp2  2
n1
n2
Chapters 8 and 9 Summary Fall 2007
Page 1 of 12
Interval Estimation
Confidence Interval for a parameter:
General Format:  Estimator  ME 
ME = (t* or z*) × SE(Estimator)
Parameter

Estimator
Estimate of the SE(Estimator)
X
S
n
d
1 n
= 1 – 2 X d   di
n i 1
(Dependent
Sd
n
Samples)
1 – 2
(Independent
Samples)
p
p1 – p2
(Independent
samples)
p1 – p2
(Dependent
Samples)
X1  X 2
S12 S22

n1 n2
p̂
ˆp( 1  ˆp )
n
ˆp1  ˆp2
ˆp1 ( 1  ˆp1 ) ˆp2 ( 1  ˆp2 )

n1
n2
ˆp1  ˆp2
Not covered
Chapters 8 and 9 Summary Fall 2007
Page 2 of 12
Hypothesis Testing
Steps in Hypothesis Testing:
1. Problem identification
(Define the population(s), parameter(s),
technique to use, etc.)
2. Check for assumptions
 Types of random variables
 Random Sampling
 Samples large enough? [Depends on 1]
3. Specify the hypotheses
 One sample test on population proportion:
i. Ho: p = po vs. Ha: p < p0 or
ii. Ho: p = po vs. Ha: p > p0 or
iii. Ho: p = po vs. Ha: p  p0
 One sample test on population mean:
i. Ho:  = 0 vs. Ha:  > 0 or
ii. Ho:  = 0 vs. Ha:  < 0 or
iii. Ho:  = 0 vs. Ha:   0
 Test for difference of population means (i.e.,
d = 1 – 2) using two dependent samples
i. Ho: d = 0 vs. Ha: d > 0 or
ii. Ho: d = 0 vs. Ha: d < 0 or
iii. Ho: d = 0 vs. Ha: d  0
Chapters 8 and 9 Summary Fall 2007
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 Test on difference of two population
proportions (using either dependent or
independent samples)
i. Ho: p1 – p2 = 0 vs. Ha: p1 – p2 > 0 or
ii. Ho: p1 – p2 = 0 vs. Ha: p1 – p2 < 0 or
iii. Ho: p1 – p2 = 0 vs. Ha: p1 – p2  0
 Test for difference of population means (i.e.,
1 – 2) using two independent samples
i. Ho: 1 – 2 = 0 vs. Ha: 1 – 2 > 0 or
ii. Ho: 1 – 2 = 0 vs. Ha: 1 – 2 < 0 or
iii. Ho: 1 – 2 = 0 vs. Ha: 1 – 2  0
4. Specify the Test Statistic to use
(Depends on the hypotheses)
General Form:
Estimator  Number in Ho
TS  Z 
~ N( 0,1)
SE( Estimator )
Estimator  Number in Ho
TS  T 
~ t( d f )
Est. SE( Estimator )
See the following table for the test statistics (and
their distributions) for testing hypotheses on
different parameters.
Chapters 8 and 9 Summary Fall 2007
Page 4 of 12
Parameter

d = 1 – 2
(Dependent Samples)
1 – 2
(Independent Samples)
Test Statistic and its Distribution
X  0
~ t( n1 )
S/ n
X  0
T d
~ t( n1 )
Sd / n
( X 1  X 2 )  0
T
~ t( df )
2
2
S1 S2

n1 n2
T
df = smaller of {(n1 – 1), (n2 – 1)}
or df = (n1 – 1) + (n2 – 1)
p
p1 – p2
(Independent samples)
p1 – p2
(Dependent Samples)
p̂  p0
~ N( 0,1 )
p0 ( 1  p0 )
n
( ˆp1  ˆp2 )  0
Z
~ N( 0,1 )
1 1
ˆp( 1  ˆp )   
 n1 n2 
YN  NY
Z
~ N( 0,1 )
YN  NY
Z
DO NOT FORGET TO VERIFY
THAT THE ASSUMPTION(S) NEEDED
FOR THE ABOVE DISTRIBUTIONS
IS/ARE SATISFIED
Chapters 8 and 9 Summary Fall 2007
Page 5 of 12
5. The p-value (Depends on the alternative)
 If Ha: p < p0 then p-value = P(Z ≤ Zcal)
 If Ha: p > p0 then p-value = P(Z ≥ Zcal)
 If Ha: p  p0 then p-value = 2×P(Z ≥ | Zcal| )
 If Ha: p1 – p2 < 0 then p-value = P(Z ≤ Zcal)
 If Ha: p1 – p2 > 0 then p-value = P(Z ≥ Zcal)
 If Ha: p1 – p2  0
then p-value = 2×P(Z ≥ | Zcal| )
 If Ha:  < 0 then p-value = P(T ≤ Tcal)
 If Ha:  > 0 then p-value = P(T ≤ Tcal)
 If Ha:   0 then p-value = 2×P(T ≥ | Tcal| )
 If Ha: d < 0 then p-value = P(T ≤ Tcal)
 If Ha: d > 0 then p-value = P(T ≥ Tcal)
 If Ha: d  0 then p-value = 2×P(T ≥ |Tcal| )
 If Ha: 1 – 2 < 0
 If Ha: 1 – 2 > 0
then p-value = P(T ≤ Tcal)
then p-value = P(T ≥ Tcal)
 If Ha: 1 – 2  0
then p-value = 2×P(T ≥ |Tcal| )
Chapters 8 and 9 Summary Fall 2007
Page 6 of 12
6. Decision
 Only 2 possible decisions:
i. Reject Ho
ii. Do not reject Ho (or Fail to reject Ho)
 Decision rule is always the same:
Reject Ho if p-value ≤ 
 That is,
i. Small p-values give evidence that
support Ha, so we reject Ho
ii. Large p-values do not give any
evidence to support Ha, so we do not
reject Ho.
7. Conclusion
 Explanation of the decision for the layman
 Should not include any technical jargon.
Chapters 8 and 9 Summary Fall 2007
Page 7 of 12
Some concepts related to statistical Inference
 Hypothesis: A statement about one or more
population parameter.
 Decisions
 Reject Ho (does not necessarily mean it is
false.)
 Do Not reject Ho (does not necessarily mean
it is true.)
 Erroneous Decisions:
o Type I Error: Rejecting Ho when in fact it
is true.
o Type II Error: NOT Rejecting Ho when
in fact it is NOT true.
 Probabilities of Erroneous Decisions
o
 = P(Type I Error) = Level of significance of a test
o (1 – )×100 = Confidence Level
o  = P(Type II Error)
o 1 –  = Power of a test
o CanNOT have both  and  small.
o Balance needed between  and .
Chapters 8 and 9 Summary Fall 2007
Page 8 of 12
Limitations (of confidence intervals and significance tests)
 Must have data from random samples (Data from
non-random samples are useless and may be
misleading, GIGO).
 Outliers may cause problems. Special care is
needed.
 “Do not reject Ho”  “Accept Ho.”
 “Reject Ho”  “Accept Ha”
 Statistical Significance  Practical significance
 P-value  P(Ho is true)
 ME takes into account ONLY sampling
variability, but NOT non-sampling errors
(measurement errors, non-coverage, nonresponse, etc.)
 CI gives more information than a significance
test: “A confidence interval is the set of all
acceptable hypotheses”
 Interpretation of CI
o Verbal explanation
o Meaning when
 (–, –) Both of ends of CI negative
 (–, +) One end negative, one positive
 (+,+) both ends positive
 Association does not mean causation
Chapters 8 and 9 Summary Fall 2007
Page 9 of 12
 Response variable is the variable of interest that
is measured
 Explanatory Variable is the variable that
explains (some of the) changes in the response
variable.
 Matched (Dependent) Samples control the effect
of lurking variables. Should be preferred when
there is a positive correlation between response
and the controlled variables.
 Experimental control (same environment) vs.
statistical control (same values) of lurking
variables.
 Problem identification is a necessary first step in
statistical inference.
 Simpson’s paradox: Be careful in interpreting
contingency tables. There may be a third factor
that influences both of the variables. Controlling
for the effect of this third variable may change the
conclusions.
Chapters 8 and 9 Summary Fall 2007
Page 10 of 12
Types of hypotheses: So far we have seen the
following hypothesis testing procedures:
I. Inference on . one population mean.
II. Inference on d = 1 – 2 using dependent
samples
III. Inference on 1 – 2 using independent samples
IV. Inference on one population proportion, p
V. Inference on p1 – p2 with dependent samples
VI. Inference on p1 – p2 using independent samples
In the test you will be given the formula you see on
the next page. You have to remember everything
else. For example, you must learn the following two
formulae. They will not be given in the test.
CI   Estimator
 ( t* or z* )  SE( Estimator )
Estimator  Number in Ho
~ N( 0,1 )
SE( Estimator )
Estimator  Number in Ho
TS  T 
~ t( df )
Est. SE( Estimator )
TS  Z 
What are the conditions under which the formulas on
the next page are true?
Chapters 8 and 9 Summary Fall 2007
Page 11 of 12
Parameter

d = 1 –
2
(Dependent
Samples)
1 – 2
(Independe
nt
Samples)
Estimator
Estimate of the SE(Estimator)
X
S/ n
Xd
Sd / n
Test Statistic and its Distribution
X  0
T
~ t( n 1 )
S/ n
T
T
X1  X2
2
1
2
2
S
S

n1 n2
ˆp( 1  ˆp )
;
n
p0 ( 1  p0 )
For Tests:
n
ˆp1 ( 1  ˆp1 ) ˆp2 ( 1  ˆp2 )
For CI:

n1
n2
p̂
p1 – p2
(Independe
nt samples)
ˆp1  ˆp2
p1 – p2
(Dependent
Samples)
ˆp1  ˆp2
For Tests:
( X 1  X 2 )  0
~ t( df )
S12 S 22

n1 n2
df = smaller of {(n1 – 1), (n2 – 1)}
or df = (n1 – 1) + (n2 – 1)
For CI:
p
X d  0
~ t( n 1 )
Sd / n
p̂  p0
Z
Z
1 1
ˆp( 1  ˆp )   
 n1 n2 
Not covered
p0 ( 1  p0 )
n
~ N( 0,1)
( ˆp1  ˆp2 )  0
1 1
ˆp( 1  ˆp )   
 n1 n2 
Z
~ N( 0,1 )
YN  NY
~ N( 0,1 )
YN  NY
You will get these formulas in the test.
Make sure you understand them and start using them
NOW so that you know what is in the table and you
how to use them and when to use them.
Make sure conditions are satisfied before you start
using them.
Chapters 8 and 9 Summary Fall 2007
Page 12 of 12