Download Anatomy: Confidence Intervals

ANATOMY OF A CONFIDENCE INTERVAL FOR m WHERE n < 30
When SIGMA, s, is KNOWN and n < 30
When SIGMA, s, is UNKNOWN and n < 30
NOTE: Variable must be normally distributed
NOTE: Variable must be approximately normally distributed
EXAMPLE: A random sample of 19 women results In a mean height of
63.85 inches. Other studies have shown that women’s heights are normally
distributed with a standard deviation of 2.5 inches. Construct a 90%
confidence interval for the mean height of all women.
EXAMPLE: In a time study, 20 randomly selected managers were found to
spend a mean time of 2.40 hrs. per day on paperwork. The standard deviation
of the scores is 1.30 hours. Construct a 98% confidence interval for the mean
time spent on paperwork by all managers.
In this example we are given a standard deviation and normality based on
prior studies. Given that sigma is known, the variable is normally distributed
and n < 30 we can use the formula:
In this example we are given a standard deviation based on a sample of n =
20. Therefore we will use the formula:
 s 
x  z 

2
 n
x  t / 2,df (
s
)
n
From the given information we have
__
__
s = 2.5
x  63.85
and confidence level = 90% or .90 which corresponds to:
z / 2  1.645
s  1.30
x  2.4
n  20
From the given information we have
n  19
__
There the value of t / 2,df is obtained from the t- distribution table for the
given confidence level which corresponds to: t
 2.539
.01,19
The 98% confidence interval for m is given by:
The 90% confidence interval for m is given by:
 s 
 2.5 
x  z 
  63.85  1.645
  63.85  .943
2 n 
 19 
(here: 63.85 - .943 = 62.907 in. and 63.85 + .943 = 64.793 in.)
x  t
2
,df
 s 
 1.3 

  2.4  2.539
  2.4  .738
 n
 20 
(here: 2.4 - .738 = 1.662 hrs. and 2.4 + .738 = 3.138 hrs.)
In Words:
We are 90% confident that the mean height of women lies between 62.907
inches and 64.793 inches.
In Words:
We are 98% confident that the mean hours spend on paperwork by managers
is between 1.662 hours and 3.138 hours.
In Notation:
The 90% confidence interval for m may be presented in any of three ways:
In Notation:
The 98% confidence interval for m may be presented in any of three ways:
(62.907  m  64.793)
or
or
(62.907,64.793)
m  63.85  .943
[inches]
(1.662  m  3.138)
or
or
(1.662,3.138)
m  2.40  .738
[hours]