Transcript 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
EXAMPLE: The National Center for Educational Statistics surveyed 4400
college graduates about the lengths of time required to earn their bachelor’s
degree. The mean was found to be 5.15 years. Prior studies have found a
standard deviation of 1.68 years. Construct a 98% confidence interval for m.
EXAMPLE: A random sample of 676 American men between the ages of 45
and 65 yielded a mean energy intake per day of 2189 calories, and a standard
deviation of 949 calories. Construct a 99% confidence interval for m.
In this example we are given a known standard deviation and the sample size
is  30. Therefore we will use the formula:
In this example we are given a standard deviation based on a sample
(population standard deviation is unknown, and n  30). Therefore we will
use the formula:
__
 s 
x  z 

2
 n
x  z / 2
From the given information we have:
n
From the given information we have:
__
__
x  5.15
n  4400
s
s  1.68
and confidence level = 98% or .98 which corresponds to:
z / 2  2.33
The 98% confidence interval for m is given by:
s  949
x  2189
n  676
and confidence level = 99% or .99 which corresponds to:
z / 2  2.575
The 99% confidence interval for m is given by:
__
 s 
 1.68 
x  z 
  5.15  2.33
  5.15  .059
2 n 
 4400 
x  z / 2
n
 2189  2.575
949
 2189  2.575
949
676
 2189  93.99  2095.01
and
__
x  z / 2
(here: 5.15 - .059 = 5.091 yrs and 5.51 + .059 = 5.209 yrs.)
s
s
n
676
 2189  93.99  2282.99
In Words:
We are 98% confident that the mean years to obtain a bachelor’s degree is
between 5.091 years and 5.209 years.
In Words:
We are 99% confident that the mean energy intake of all American men
between the ages of 45 and 65 is between 2095.01 and 2282.99 calories
In Notation:
The 98% confidence interval for m may be presented in any of three ways:
In Notation:
The 99% confidence interval for m may be presented in any of three ways:
(5.091  m  5.209)
or
or
(5.091,5.209)
m  5.15  .059
[years]
(2095.01  m  2282.99)
or
or
(2095.01,2282.99)
m  2189.0  93.99
[calories]