Transcript Populations - State University of New York at Oswego

Choosing the Sample Size
Confidence Interval for a Mean
Given
A random sample of size n from a Normal
population with mean . (n/N  0.05)
Result
A confidence interval is given by
X E
where
E  t*
S
n
where t* is the appropriate critical value for the
T distribution with (n – 1) DF.
Example
A radar gun was used to find the speed of 19
vehicles passing a checkpoint in a 30 mph zone
near a school. The distribution of speeds was
unimodal and symmetric, with mean 32.83 and
standard deviation 6.292. The error margin for a
90% confidence interval is.
E  1 . 734
6 . 292
19
 2 . 50
Example
Suppose we want to estimate the mean speed of
vehicles with 90% confidence and error margin
no greater than  0.5 mph.
E  t*
S
with ( n  1) DF
n
Results of the prior study have the standard
deviation around 6.3. (There is no guarantee that
S will be 6.292 for a future study. In fact: it
won’t. You can fudge this, but if you fudge small
and the actual result is larger, you will miss your
goal.)
Solution: Trial and Error
With n = 19, error margin = 2.5. To reduce the
error margin by 5, try increasing sample size
5: 5(19) = 95…
If n = 95 then DF = 94 and t* = 1.661.
(Notice that z* = 1.645 – quite close.)
E  1 . 661
6 .3
95
 1 . 07
A 5 increase in n doesn’t produce a 5 decrease
in E. More like a square root of 5 = 2.24
decrease: 2.5/2.24 = 1.12.
Solution: Trial and Error
n = 95 gives about 1.07 for the error margin. We
want to cut that in two. So: quadruple (4 = 22)
the sample size.
If n = 380 then DF = 379 and t* = 1.649.
(Notice that z* = 1.645 – very very close.)
E  1 . 649
6 .3
 0 . 533
380
A few more than 380 will do the trick.
Solution: Via Formula
Since the t* value will be close to the z* value
(unless n is very small), we can use z* to
determine the sample size.
 z*S 
n

 E 
2
Solution: Example
With confidence 95%, z* = 1.96.
We’re taking S = 6.3 (although this is only an
educated guess)
The desired error margin is E = 0.5 mph.
2
2
 z*S 
 1 . 645  6 . 5 
n
 
  429 . 61
0 .5
 E 


Sample at least n = 430 vehicles.
What Do I Use for S?
Previously collected data – even on a similar but not identical
variable – is useful.
The “Range Rule of Thumb” can be useful in generating an
educated guess for the standard deviation.
To ensure the error margin, guess conservatively (too high).
If you are too conservative, you will oversample (costs money, wastes
time)
If the guess is too small (S actually turns out larger than what you
guessed) you will not achieve the target for E
If the sample size you obtain is small (say less than 30), you may
want to increase it by a bit to adjust for the difference between t*
and z*.