Size
Sample
Population
IME 301
n
N
Mean

x

i n
Variance
x
i 1 i
n


i n
S
2
i 1


n 1
i N

2
i 1
( xi  x )
2
( xi   )
N
2
b = is a random value
 = is probability
z  b
means
P ( z  b)  1  
For example:
z0.025  b
means P ( z  b)  1  0.025  0.975
Then from standard normal table: b = 1.96
Also:
(b)  P( Z  b)
For example
(1.96)  P( Z  1.96)  0.975
IME 301
• Point estimator and Unbiased estimator
• Confidence Interval (CI) for an unknown parameter
is an interval that contains a set of plausible values of
the parameter. It is associated with a Confidence
Level (usually 90% =<CL=< 99%) , which measures
the probability that the confidence interval actually
contains the unknown parameter value.
_
_
CI = X – half width, X + half width
Z / 2
An example of half width is:
n
• CI length increases as the CL increases.
• CI length decreases as sample size, n, increases.
• Significance level (
= 1 – CL)

IME 301
Confidence Interval for Population Mean
Two-sided, t-Interval
Assume a sample of size n is collected. Then
sample mean, X ,and sample standard deviation, S,
is calculated.
The confidence interval is:
X
IME 301 (new Oct 06)
t / 2,( n1) S
n
X 
t / 2,( n1) S
n
• Interval length is:
L  2*
t / 2,( n1) S
n
• Half-width length is:
half  width 
t / 2,( n1) S
n
• Critical Points are:
X
IME 301
t / 2,( n1) S
n
and
X
t / 2,( n1) S
n
Confidence Interval for Population Mean
One-sided, t-Interval
Assume a sample of size n is collected. Then
sample mean, ,and sample standard deviation, S,
X
is calculated.
The confidence interval is:
t ,( n1) S OR
    X 
n
IME 301 new Oct 06
X
t ,( n1) S
n
   
Hypothesis: Statement about a parameter
Hypothesis testing: decision making
procedure about the hypothesis
Null hypothesis: the main hypothesis H0
Alternative hypothesis: not H0 , H1 , HA
Two-sided alternative hypothesis, uses 
One-sided alternative hypothesis, uses > or <
IME 301
Hypothesis Testing Process:
1. Read statement of the problem carefully (*)
2. Decide on “hypothesis statement”, that is H0 and HA (**)
3. Check for situations such as:
normal distribution, central limit theorem,
variance known/unknown, …
4. Usually significance level is given (or confidence level)
5. Calculate “test statistics” such as: Z0, t0 , ….
6. Calculate “critical limits” such as: Z  / 2 t / 2 ,( n1)
7. Compare “test statistics” with “critical limit”
8. Conclude “accept or reject H0”
IME 301
FACT
H0 is true
Accept
H0
no error
H0 is false
Type II
error
Decision
Reject
H0
Type I
error
no error

=Prob(Type I error) = significance level
= P(reject H0 | H0 is true)

= Prob(Type II error) =P(accept H0 | H0 is false)
(1 -
 ) = power of the test
IME 301
The P-value is the smallest level of significance that
would lead to rejection of the null hypothesis.
The application of P-values for decision making:
Use test-statistics from hypothesis testing to find Pvalue. Compare level of significance with P-value.
P-value < 0.01 generally leads to rejection of H0
P-value > 0.1 generally leads to acceptance of H0
0.01 < P-value < 0.1 need to have significance level to
make a decision
IME 301 (new Oct 06)
Test of hypothesis on mean, two-sided
No information on population distribution
H 0 :   0
H1 :    0
n ( X  0 )
s
Test statistic:
t0 
Reject H0 if
 t / 2,( n 1)  t0 ..or..t0  t / 2,( n 1)
or P-value = 2 * P(tn1  t0 )  
IME 301
Test of hypothesis on mean, one-sided
No information on population distribution
H 0 :   0
H 0 :   0
H A :   0
Test
statistic:
Reject Ho if
P-value =
OR
Reject H0
if
IME 301
H A :   0
n ( X  0 )
t
s
P(tn1  t )  
t  t ,( n 1)
P(tn1  t )  
t  t ,( n 1)
Test of hypothesis on mean, two-sided, variance known
population is normal or conditions for central limit theorem holds
H 0 :   0
H A :   0
n ( X  0 )
Test statistic:
Z0 
Reject H0 if
 Z / 2  Z 0 ..OR..Z 0  Z / 2
or, p-value =
IME 301
0

2[1  (| Z 0 |)]  2 * ( | Z 0 |)  
Test of hypothesis on mean, one-sided, variance known
population is normal or conditions for central limit theorem holds
H 0 :   0
H 0 :   0
H A :   0
Test statistic:
Reject Ho if
P-value =
Or,
Reject H0 if
IME 301 and 312
Z0 
H A :   0
n ( X  0 )
(1  (Z 0 ))  
Z 0  Z

( Z 0 )  
Z 0   Z
Type II error, for mean with known variance

 n
   Z / 2 




 n
    Z / 2 









    0
Where
Sample size, for mean with known variance
 Z
n


 /2
 Z

 

Where
IME 301, Feb. 99
Two-sided


 Z  Z   
n


2


2
One-sided
  ( Z  )