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When we free ourselves of desire,
we will know serenity and freedom.
1
Inferences about One Population
Mean
Estimation:
z interval
and t interval
Statistical tests: z test
and t test
2
Estimation
To estimate a numerical summary in population
(parameter):


3
Point estimator — A (same) numerical
summary in sample; a statistic
Interval estimator — a “random” interval
which includes the parameter most of time
Example: Serum Cholesterol Level





4
Population: all 20- to 74-year-old male
smokers in the U.S.
X: serum cholesterol level
Distribution: normal with unknown mean m
Sample: 12 randomly selected male smokers
with mean x  217 mg / 100ml
Question: what is the mean serum cholesterol
level of adults U.S. male smokers?
Confidence Interval
ˆL  ˆL (Y1 ,..., Yn )
ˆL  ˆL (Y1 ,..., Yn )
ˆL  ˆL ( X1 ,..., X n )
ˆU  ˆU ( X1 ,..., X n )
5
Ideally, a short interval with high confidence interval is preferred.
6
Estimation for m


Point estimator: x
Confidence interval
–
Normality with known s or large sample (n > 30) :
Z interval
x  z / 2
–
n
Normality with unknown s: t interval
x  t / 2
7
s
s
n
The t distribution

The distribution of t, the standardized sample
mean with estimated standard deviation for a
normal population
xm
t
s/ n
8
9
One Population
10
Sample Size for Estimating m
( z / 2 ) s
n
2
E
2
2
if s is unknown,
use s from prior
data or the upper
bound of s.
Where E is the largest tolerable error.
11
Hypothesis Test

The null hypothesis:
H 0 : m  m0  211mg / 100ml

The alternative hypothesis:
H a : m  211mg / 100ml
Question: Should we reject or not reject Ho?
12
The Logic of Hypothesis Test
“Assume the null hypothesis Ho is a
(possible) truth until proven false”
Analogical to
“Presumed innocent until proven guilty”
The logic of the US judicial system
13
Rejection Region



14
When the observed test statistic is too
“extreme” under Ho (i.e. in the opposite
direction of Ho), the Ho should be rejected
Rejection Region is the region when the
observed test statistic falls in, we will reject Ho
A test result is determined by its rejection
region, while rejection region is determined by
the significance level
Types of Errors
H0 true
we accept H0
we reject H0
Good!
(Correct!)
Type I
Error, or
“ Error”
H0 false
Type II
Error, or
“ Error”
Good!
(Correct)
15
Terms




16
 Type I error rate = probability of
incorrectly rejecting Ho
 Type II error rate = probability of
incorrectly accepting Ho
Significance level is the pre-set largest
tolerable  level
Power = probability of correctly rejecting
Ho = 1-
Z Test Statistic
For normal populations or large samples (n > 30):
Z
x  m0
sx
x  m0

s/ n
And the computed value of Z is denoted by Z*.
17
Types of Tests
18
Types of Tests
19
Types of Tests
20
Example: Serum Cholesterol Level
21

Assuming the population S.D. is 46
mg/100ml, conduct the test at 5 % level

What is the power of the test at the true
m221 mg/100ml?
22
Steps in Hypothesis Test
1.
2.
3.
4.
5.
23
Set up the null (Ho) and alternative (Ha)
hypotheses
Find an appropriate test statistic (T.S.)
Find the rejection region (R.R.)
Reject Ho if the observed test statistic falls in
R.R.
Report the result in the context of the
situation
t Test Statistic

For normal populations with unknown s
x  m0
t
s/ n
the same formula for Z but replacing s by s
24
One Population
25
Sample Size for Testing m
The type I, II error rates are controlled
at ,  respectively and the maximum tolerable error is :
( z  z  ) s
2
One-tailed tests:
n

2
( z / 2  z  ) s
2
Two-tailed tests:
26
n
2

2
2
P-value




27
p-value is the probability of seeing what we observe as
far as (or further) from Ho (in the direction of Ha) given
Ho is true; the smallest  level to reject Ho
p-value is computed by assuming Ho is true and then
determining the probability of a result as extreme (or
more extreme) as the observed test statistic in the
direction of the Ha.
The smaller p-value is, the less likely that what we
observe will occur given Ho is true.
Smaller p-value means stronger evidence against Ho.
Computing the p-Value for the Z-Test
28
Computing the p-Value for the Z-Test
29
Computing the p-Value for the Z-Test
P-value = P(|Z| > |z*| )= 2 x P(Z > |z*|)
30
Hypothesis Test using p-Value
1.
2.
3.
4.
5.
31
Set up the null (Ho) and alternative (Ha)
hypotheses
Find an appropriate test statistic (T.S.)
Find the p-value
Reject Ho if the p-value < 
Report the result in the context of the
situation
Example: Serum Cholesterol Level

32
Assuming the population S.D. is 46
mg/100ml, conduct the test at 5 % level
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