learning about statistical Inference.ppt

STATISTICAL INFERENCE
• The process of drawing
inferences or making
generalizations about
characteristics of a population
based on partial and incomplete
information obtained from a
sample of the population.

THE BASIS FOR STATISTICAL
INFERENCE
Population of
unknown
parameters
Known
Sample
The observed sample
statistics are generalized to the
unknown population
A sample is selected
from a population with
unknown parameters.

TWO MAJOR METHODS OF
STATISTICAL INFERENCE
1.Estimation of Parameters
1.1 Point Estimation
1.2 Interval Estimation
2. Hypothesis Testing
2.1 Parametric Tests
2.2 Nonparametric Tests

• the process by which sample
information is used to predict or
estimate the numerical value of some
population measure. The formula,
function or procedure used in
estimating a population parameter is
called an estimator.
ESTIMATION OF PARAMETERS

Two kinds of Parameter Estimates
1. Point estimate
- consists of a single value used to
estimate population parameter.
2. Interval estimate
- consists of a range of values
- also confidence intervals.

Formula for the Confidence Interval of
the Mean for Specific α
















n
z
X
n
z
X




THE COMMONLY USED CONFIDENCE
LEVELS

Example
• A researcher wishes to estimate the
average amount of money a person spends
on lottery tickets each month. A sample of
50 people who play the lottery found the
mean to be $19 and the standard deviation
to be 6.8. Find the best point estimate of
the population mean and the 95%
confidence interval of the population mean.

learning about statistical Inference.ppt

INTERPRETING THE CONFIDENCE INTERVAL ESTIMATE
n
z
x


This notation means that, if we repeatedly
draw samples of size 50 people from this
population , 95% of the values of sample mean
( ) will be such that population mean ( )
would lie somewhere between 19 1.885,
and 5% of the value of mean will produce
intervals that would not include population
mean.
x 


Sample Size
• The formula for sample size is derived from
the maximum error of estimate formula
estimate
of
error
maximum
the
is







n
z
E

 

z
n
E 
2





 

E
z
n


SAMPLE SIZE
The college president asks the statistics
teacher to estimate the average age of the
students at their college. How large a
sample size is necessary? The statistics
teacher would like to be 99% confident that
the estimate should be accurate within one
year. From a previous study, the standard
deviation is known to be 3 years.

CONFIDENCE INTERVALS FOR PROPORTIONS
n
q
p
z
p
p
n
q
p
z
p
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ 



size
sample
and
interest
of
stics
characteri
the
possess
that
units
sample
of
number
where
ˆ
1
or
ˆ
and
ˆ
,
proportion
sample
a
For
proportion
sample
ˆ
proportion
population








n
x
p
n
x
n
q
n
x
p
p
p
When np and nq are each greater than or
equal to 5.

EXAMPLE
A sample of 500 nursing applications
included 60 from men. Find the 90%
confidence interval of the true
proportion of men who applied to the
nursing program.

FORMULA FOR MINIMUM SAMPLE SIZE NEEDED
FOR INTERVAL ESTIMATE OF A POPULATION
PROPORTION
2
ˆ
ˆ 






E
z
q
p
n

EXAMPLE
A researcher wishes to estimate, with 95%
confidence, the proportion of people who
own a home computer. A previous study
shows that 40% of those interviewed had a
computer at home. The researcher wishes
to be accurate within 2% of the true
proportion. Find the minimum sample size
necessary.

QUIZ
1. The proportion of students in private schools is around
11%. A random sample of 450 students from a wide
geographic area indicated that 55 attended private
schools. Estimate the true proportion of students
attending private schools with 95% confidence.
2. A researcher wishes to estimate the proportion
of adult males who are under 5 feet 5 inches tall.
She wants to be 90% confident that her estimate
is within 5% of the true proportion. How large a
sample should be taken if in a sample of 300
males, 30 were under 5 feet 5 inches tall?

CONFIDENCE INTERVALS FOR THE
MEAN (σ UNKNOWN AND n < 30)
















n
s
t
X
n
s
t
X 
FORMULA

CONFIDENCE INTERVALS FOR THE
MEAN (σ UNKNOWN AND n < 30
The data represent a sample of the number of
home fires started by candles for the past
several years. Find the 99% confidence
interval for the mean number of home fires
started by candles each year
5460 5900 6090 6310
7160 8440 9930
3
.
1610
4
.
7041


s
x

HYPOTHESIS -TESTING
• refers to the procedures for
establishing a set of rules that
lead to the acceptance or
rejection of a statement or
hypothesis about a population.

THE LOGIC OF
HYPOTHESIS TESTING
• Evidences from the sample that are
inconsistent with the stated hypothesis lead
to rejection of the hypothesis while evidence
supporting the hypothesis lead to its
acceptance.
• Rejection of HO is to conclude that it is false
while its acceptance is a result of insufficient
evidence to reject it and does not necessarily
imply that it is true.

THE GOAL OF HYPOTHESIS
TESTING
• To arrive at a decision to accept
or reject the null hypothesis on
the basis of a computed test
statistic based on values
obtained from the sample data.

ULTIMATE GOAL OF STATISTICAL
INFERENCE
• To be able to draw valid
inferences or generalizations
concerning the population under
study based on evidence from
observed samples.

HYPOTHESIS TESTING
• The null hypothesis, symbolized by H0, is a
statistical hypothesis that states that there is no
difference between a parameter and a specific
value, or that there is no difference between two
parameters.
• The alternative hypothesis (or research
hypothesis), symbolized by H1, is a statistical
hypothesis that states the existence of a
difference between parameter and a specific
value, or states that there is a difference between
two parameters.

H0 : The defendant is innocent
H1 : The defendant is guilty

Type I Type II Error
DECISION
Ho is true
(Defendant is Innocent)
Ho is false
(Defendant is Guilty)
Reject Ho
Convict
defendant
ERROR
Type I
P(Type 1 Error) = α
CORRECT DECISION
Do not Reject
Ho
Acquit the
defendant
CORRECT DECISION
ERROR
Type II β
P(Type II Error) = β

The critical concepts in hypothesis testing follow.
1.There are two hypotheses. One is called the null hypothesis, and
the other the alternative or research hypothesis.
2.The testing procedure begins with the assumption that the null
hypothesis is true.
3.The goal of the process is to determine whether there is enough
evidence to infer that the alternative hypothesis is true
4.There two possible decisions:
Conclude that there is enough evidence to support the alternative
hypothesis
Conclude that there is not enough evidence to support the
alternative hypothesis
Two possible errors can be made in any test. A type I error occurs
when we reject a true null hypothesis, and a Type II error occurs
when we don’t reject a false null hypothesis.

HYPOTHESIS TESTING
Two-tailed test Right-tailed test Left-tailed
test
k
H 

:
0
k
H 

:
1
k
H 

:
0
k
H 

:
1
k
H 

:
0
k
H 

:
1
k
H 

:
0

HYPOTHESIS TESTING
Ex. State the null and alternative hypotheses for
each conjecture.
a. A researcher thinks that if expectant mothers use
vitamin pills, the birth weight of the babies will
increase. The average birth weight of the population is
8.6 pounds.
b. An engineer hypothesize that the mean number of
defects can be decreased in a manufacturing process
of compact disks by using robots instead of humans for
certain tasks. The mean number of defective disks per
1000 is 18.

HYPOTHESIS TESTING
c. A psychologist feels that playing soft music during
a test will change the results of the test. The
psychologist is not sure whether the grades will be
higher or lower. In the past, the mean of the
scores was 73.
A statistical test uses the data obtained from a
sample to make a decision about whether the null
hypothesis should be rejected.
The numerical value obtained from statistical test
called the test value.

pounds
H 6
.
8
:
0


pounds
H 6
.
8
:
1


18
:
0


H
k
H 

:
1
73
:
0


H
73
:
1


H

HYPOTHESIS TESTING
• A type I error occurs if one rejects the null
hypothesis when it is true.
• A type II error occurs if one does not reject
the null hypothesis when it is false.
• The level of significance is the maximum
probability of committing a type I error. This
probability is symbolized by (Greek letter
alpha). That is,



error)
I
type
(
P

HYPOTHESIS TESTING
• Solving Hypothesis-Testing Problems (Traditional
Method)
Step 1State the hypotheses and identify the claim.
Step 2Find the critical value(s). p – value
Step 3Compute the test value.
Step 4Make the decision to reject or not reject the null
hypothesis.
Step 5Summarize the results.

• “The mark of maturity is the capacity for
reflective commitment. This involves, on one
hand, full recognition of the funded wisdom of
the race, full respect for all the knowledge
available to man and relevant to any specific
decision. It also involves the recognition that
knowledge is never complete, that all evidence is
never in, and that we must, of necessity, decide
and act on the basis of partial information and
take the risk of being partially wrong.”
-Theodore M. Greene

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