10.1. Population Mean: Large Sample Case ( n  30 )
Motivating example:
In the survey conducted by CJW, Inc., a mail-order firm, the satisfaction scores
(1~100) of 100 customers (n=100) are obtained. Suppose   20 is known. Also,
x  82,  x 

n

20
100
 2.
Derivation of 95% confidence interval:
Since
X  N (  ,  X2 ) .
Thus,
X 
X
 Z (~ N (0,1))
.
Then,
 X 

 X

0.95  P Z  1.96  P
 1.96   P
 1.96 
 

 

 X

 X



X
 P  1.96 
 1.96   P 1.96 X    X  1.96 X 
X


 PX  1.96 X    X  1.96 X 
 There is an approximate 95% chance that the population mean  will fall
X  1.96 X
between
and
X  1.96 X
Pu  X  1.96 X , X  1.96 X   0.95 ,
X  1.96
X
, X  1.96 X  with a chance close to 0.95.
Note: in the above equation,
1
,
 falls in the interval
i.e.,
 X 


0.95  P
 1.96   P X    1.96 X
 

X




There is an approximate 95% chance that the sample mean will
provide a sampling error of
1.96 X .
Example (continue)
In the above example, since  X  2 , thus
P( X  1.96 * 2    X  1.96 * 2)  PX  3.92    X  3.92  0.95
There is an approximate 95% chance that the population mean  will fall between
X  3.92
and
X  3.92 .
95% confidence interval:
Suppose the sample size is large.
 As  is known,
x  1.96 X  x  1.96
 

 
  x  1.96
, x  1.96 
n 
n
n
is a 95% confidence interval estimate of the population mean
.
 As  is unknown,
x  1.96s X  x  1.96
s 
s
s 
  x  1.96
, x  1.96 
n 
n
n
is a 95% confidence interval estimate of the population mean
where
s2
is the sample variance and s X is the estimate of  X .
Example (continue)
In the above example, since x  82 ,  X  2 , and n  100 , thus
x  1.96

20
 82  1.96
 82  3.92  78.08, 85.92
n
100
2
,
.
is a 95% confidence interval estimate of the population mean
General confidence interval:
Definition of
z
2
:
Let Z be the standard normal random variable. Then,

P Z  z   .
2

2
As   0.5 ,
P Z  z   1   .
2

 z
z
2
2
Example:
P Z  1.64  0.9  1  0.1  P Z  z 0.1   1  0.1  z 0.1  z 0.05  1.64
2
2

P Z  1.96  0.95  1  0.05  P Z  z 0.05   1  0.05  z 0.05  z 0.025  1.96
2
2

P Z  2.576  0.99  1  0.01  P Z  z 0.01   1  0.01  z 0.01  z 0.005  2.576
2
2

In summary,

0.1
0.05
0.01
1

2
0.9
0.95
0.99
0.05
0.025
0.005
3
z
2
z 0.05  1.64
z 0.025  1.96
z 0.005  2.576
Derivation of 1    100% confidence interval:
As the sample size is large,
 X 

 X

1    P Z  z   P
 z   P 
 z 
 
 
2
2
2

 X

 X



X
 P  z 
 z   P  z  X    X  z  X 
2
2
2
2


X


 P X  z  X    X  z  X 
2
2


 There is an approximate 1     100% chance that the population mean 
will
fall
between
X  z  X
2
and
P u   X  z  X , X  z  X    1   ,

 
2
2

X  z  X
2
,
i.e.,
 falls in the interval
 X  z  , X  z   with a chance close to 1   .


X
X 

2
2

Note: as   0.05 , the above derivations are exactly the same as the
ones for 95% confidence interval estimate.
Motivating Example (continue)
As   0.1,
P( X  z 0.1 *  X    X  z 0.1 * X )  PX  z 0.05 * 2    X  z 0.05 * 2
2
2
 PX  3.28    X  3.28  1  0.1  0.9
There is an approximate 90% chance that the population mean  will fall between
X  3.28
and
X  3.28 .
1    100% confidence interval:
Suppose the sample size is large.
 As  is known,
4
x  z  X  x  z
2
2
 

 
  x  z
, x  z

2
2
n 
n
n
is a 1    100% confidence interval estimate of the population
mean
.
 As  is unknown,
x  z s X  x  z
2
2
s 
s
s 
  x  z
, x  z

2
2
n 
n
n
is a 1    100% confidence interval estimate of the population mean
.
Motivating Example (continue)
As   0.1,
x  z
2

20
20
 82  z 0.05
 82  1.64
 78.72, 85.28
n
100
100
is a 90% confidence interval estimate of the population mean
.
Note: in the CJW, Inc. example, the 95% confidence interval is
wider than the 90% confidence interval. Intuitively, if we want to
make sure that we will make less mistakes, we should speak vaguely
(wider confidence interval). For instance, if we want to get a 100%
confidence interval (for sure), the interval  ,  would make us
not make any mistake.
Note: the length of the confidence interval is
Therefore, a larger sample size
n
and a greater precision.
5
2 z

2
n
or 2 z
s
2
n
.
will provide a narrow interval
Example A:
A random sample of 81 workers at a company showed that they work an average of
100 hours per month with a standard deviation of 27 hours. Compute a 95%
confidence interval for the mean hours per month all workers at the company work.
[solution:]
As   0.05 ,
x  z
2

27
27
 100  z0.025
 100  1.96   94.12, 105.88
9
n
81
is a 95% confidence interval estimate of the population mean
Online Exercise:
Exercise 10.1.1
Exercise 10.1.2
6
.