One Sample Mean Inference
(Chapter 5)
In Unit 2 we will discuss:
• How to estimate the mean from a normal distribution
and compute a 95% confidence interval for the true
unknown population mean.
• How to statistically test whether the population
sampled could have a specified population mean.
• The five parts of a statistical test.
• Type I and Type II errors associated with testing.
• In general how sample size is determined.
• Inferences about the median.
One Sample Inf-1
Inferences
about the
population
mean, 
Population
Sample
Parameters

Estimates

y s
Central
Limit
Theorem
Test a
Hypothesis
Decision
Sampling Distribution of
Interval Estimate of  and .
y s
One Sample Inf-2
Estimation of the Population Mean, .
Point Estimate of  is the sample mean:
y
The Central Limit Theorems
tell us the shape of the
sampling
distribution
95% of sample means
expected to lie
in this interval.
  
N  , 
n

2.5% of sample
means expected to
lie in this interval.
2.5% of sample
means expected to
lie in this interval.
 -1.96

n

 +1.96

n
y 

n
Interval Estimate of  is the confidence interval:
One Sample Inf-3
Interval Estimate of the Population Mean
The event
y
y in (  1.96 y ,   1.96 y )

 1.96 y
-4
-3
-2
-1
0
 1.96 y
1
2
3
4
x
y  1.96 y
 1.96 y
y
y  1.96 y
 1.96 y

is equivalent to the event
-4
-3
-2
-1
0
1
x
y  1.96 y
y
y  1.96 y
2
3
4
 in ( y  1.96 y , y  1.96 y )
One Sample Inf-4
Meaning of a 95% Confidence Interval
y
In repeated sampling, the interval would
encompass the true parameter value 95%
of the time.

-4
-3
-2
-1
0
1
2
3
4
x

y  1.96 y
y
y  1.96 y
5% of the intervals don’t include the true mean.
95% of the intervals include the true mean.
One Sample Inf-5
Critical Value for Confidence Interval
95% Confidence Interval
y  1.96 y
99% Confidence Interval
y  2.58 y
(1-α)100% Confidence Interval
y  z  y
y 

n
2
• The /2 critical value of the standard
normal distribution.
• The value of the standard normal
distribution for which P(z>z /2 )= /2 .
Read from Table 1 in Ott
One Sample Inf-6
Common Critical Values
Confidence
Coefficient
(1-)
.90
Area to left of
z-value
1-/2
.95
Area to right of
z-value
/2
.05
Corresponding
z-value
z/2
1.645
.95
.975
.025
1.96
.98
.99
.01
2.33
.99
.995
.005
2.58
.99
One Sample Inf-7
Using an Estimate of 
What if we don’t know the value of  ?
We simply replace  with its estimate s
in the confidence interval equation.
s
1 n
( yi  y ) 2

n  1 i 1
However, the resulting sampling distribution is no longer normal, but
follows a t distribution instead. This involves replacing the normal
critical value, z/2, with a critical value from a t distribution, t/2,n-1
(more on this later).
If n is large (30 or more), it turns out that because of the Central
Limit Theorem the normal is a good enough approximation to the
t, and we can therefore use z/2, after all.
One Sample Inf-8
Example
n=50
y=105 s=11
95% CI for 
99% CI for 
  0.05
  0.025
2
  0.01
  0.005
2
z  2.58
z  196
.
2
2
y  z / 2 
n
or
y  z / 2 s
105  (196
. )(
n
11
n
(10195
. , 108.05)
y  z / 2 
)
In a large amount of repeated
sampling, 95 out of 100 intervals
constructed this way would include the
true mean.
n
or
y  z / 2 s
105  ( 2.58)(
n
11
n
(100.98, 109.01)
)
In a large amount of repeated
sampling, 99 out of 100 intervals
constructed this way would include the
true mean.
One Sample Inf-9
Sample Size Determination
The CI equation can be turned around to allow computation of the
sample size needed to estimate the true mean with a specified
level of precision and confidence.
Typical Confidence Levels
90%
95%
99%
Need three things:
1) Target level of confidence: (1-)100%
2) Tolerable margin of error (half of CI width): E
3) Estimate of population standard deviation:

i) Prior Knowledge
ii) Pilot Sample
iii) Expert Guess (range/4)
Target Precision
One Sample Inf-10
100(1-)% CI for 
y  z / 2

n
Equate this to E.
Equate the CI precision to the target precision and solve for n.
z
/2

 E 
n



z / 2 E  n   z / 2 E   n
Precision can be specified relative to .
2
voilá
One Sample Inf-11
Sample Size Example
  2 oz
E   0.25
1   100%  95%

2
 .025
.
z / 2  196
 1.962 
2



4

2

1
.
96
 246 (round to next highest integer)


 .25 
2
2


 z / 2   n
E

Take home message: If you have some idea of
the underlying variability in the population and
you know roughly how precise you wish the
mean estimate to be you can compute the
required sample size.
One Sample Inf-12