Author(s): Brenda Gunderson, Ph.D., 2011
License: Unless otherwise noted, this material is made available under the
terms of the Creative Commons Attribution–Non-commercial–Share
Alike 3.0 License: http://creativecommons.org/licenses/by-nc-sa/3.0/
We have reviewed this material in accordance with U.S. Copyright Law and have tried to maximize your
ability to use, share, and adapt it. The citation key on the following slide provides information about how you
may share and adapt this material.
Copyright holders of content included in this material should contact open.michigan@umich.edu with any
questions, corrections, or clarification regarding the use of content.
For more information about how to cite these materials visit http://open.umich.edu/education/about/terms-of-use.
Any medical information in this material is intended to inform and educate and is not a tool for self-diagnosis
or a replacement for medical evaluation, advice, diagnosis or treatment by a healthcare professional. Please
speak to your physician if you have questions about your medical condition.
Viewer discretion is advised: Some medical content is graphic and may not be suitable for all viewers.
Some material sourced from:
Mind on Statistics
Utts/Heckard, 3rd Edition, Duxbury, 2006
Text Only: ISBN 0495667161
Bundled version: ISBN 1111978301
Material from this publication used with permission.
Attribution Key
for more information see: http://open.umich.edu/wiki/AttributionPolicy
Use + Share + Adapt
{ Content the copyright holder, author, or law permits you to use, share and adapt. }
Public Domain – Government: Works that are produced by the U.S. Government. (17 USC §
105)
Public Domain – Expired: Works that are no longer protected due to an expired copyright term.
Public Domain – Self Dedicated: Works that a copyright holder has dedicated to the public domain.
Creative Commons – Zero Waiver
Creative Commons – Attribution License
Creative Commons – Attribution Share Alike License
Creative Commons – Attribution Noncommercial License
Creative Commons – Attribution Noncommercial Share Alike License
GNU – Free Documentation License
Make Your Own Assessment
{ Content Open.Michigan believes can be used, shared, and adapted because it is ineligible for copyright. }
Public Domain – Ineligible: Works that are ineligible for copyright protection in the U.S. (17 USC § 102(b)) *laws in
your jurisdiction may differ
{ Content Open.Michigan has used under a Fair Use determination. }
Fair Use: Use of works that is determined to be Fair consistent with the U.S. Copyright Act. (17 USC § 107) *laws in your
jurisdiction may differ
Our determination DOES NOT mean that all uses of this 3rd-party content are Fair Uses and we DO NOT guarantee that
your use of the content is Fair.
To use this content you should do your own independent analysis to determine whether or not your use will be Fair.
Stat 250 Gunderson Lecture Notes
Learning about a Population Mean
Part 2: Confidence Interval for a Population Mean
Chapter 11: Sections 1 and 2, CI Module 3
Do not put faith in what statistics say until you have carefully considered
what they do not say.
--William W. Watt
11.1 Introduction to Confidence Intervals for Means
Chapter 10 introduced us to confidence intervals for estimating a population proportion and
the difference between two population proportions. Recall it is important to understand how
to interpret an interval and how to interpret what the confidence level really means.

The interval provides a range of reasonable values for the parameter with an associated
high level of confidence. For example we can say, “We are 95% confident that the
proportion of Americans who do not get enough sleep at night is somewhere between
0.325 to 0.395, based on a random sample of n = 935 American adults.

The 95% confidence level describes our confidence in the procedure we used to make the
interval. If we repeated the procedure many times, we would expect about 95% of the
intervals to contain the population parameter.
11.2 CI Module 3: Confidence Interval for a Population Mean 
Consider a study on the design of a highway sign. A question of interest is: What is the mean
maximum distance at which drivers are able to read the sign?
Data: A highway safety researcher will take a random sample of n = 16 drivers and measure the
maximum distances (in feet) at which each can read the sign.
Population parameter:
 = _______________ mean maximum distance to read the sign for _________________
Sample estimate
x = _______________ mean maximum distance to read the sign for _________________
But we know the sample estimate x may not equal  , in fact, the possible x values vary from
sample to sample. Because the sample mean is computed from a random sample, then it is a
random variable, with a probability distribution.
115
Sampling Distribution of the sample mean
If x is the sample mean for a random sample of size n from a population with a normal model,
then the distribution of the sample mean is:
Central Limit Theorem
If x is the sample mean for a random sample of size n from a population with any model,
with mean,  , and standard deviation  , then when n is large,
then the sampling distribution of the sample mean is approximately:
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.
Used with permission.
So the possible x values vary normally around  with a standard deviation of  . The
n
standard deviation of the sample mean,

n
, is roughly the average distance of the possible
sample mean values from the population mean  . Since we don’t know the population
standard deviation  we will use the sample standard deviation s, resulting in the standard
error of the sample mean.
Standard Error of the Sample mean
s.e.( x ) =
where s = sample standard deviation
The standard error of x estimates, roughly, the average distance of the possible x values
from  The possible x values result from considering all possible random samples of the same
size n from the same population.
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.
Used with permission.
So we have our estimate of the population mean, the sample mean x , and we have its
standard error. To make our confidence interval, we need to know the multiplier.
Sample Estimate  Multiplier x Standard error
The multiplier for a confidence interval for the population mean is denoted by t*, which is the
value in a Student’s t distribution with df = n – 1 such that the area between –t and t equals
116
the desired confidence level. The value of t* will be found using Table A.2. First let’s give the
formal result.
One-sample t Confidence Interval for 
x  t *s.e.( x )
where t * is an appropriate value for a t(n – 1) distribution.
This interval requires we have a random sample from a normal population. If the sample size
is large (n > 30), the assumption of normality is not so crucial and the result is approximate.
Important items:
 be sure to check the conditions
 know how to interpret the confidence interval
 be able to explain what the confidence level of say 95% really means
Try It! Using Table A.2 to find t*
(a) Find t * for a 90% confidence interval based on n = 12 observations.
(b) Find t * for a 95% confidence interval based on n = 30 observations.
(c) Find t * for a 95% confidence interval based on n = 54 observations.
(d) What happens to the value of t * as the sample size (and thus the degrees of freedom) gets
larger?
117
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.
Used with permission.
118
Try It! Confidence Interval for the Mean Maximum Distance
Recall the study on the design of a highway sign. The researcher wanted to learn about the
mean maximum distance at which drivers are able to read the sign. The researcher took a
random sample of n = 16 drivers and measured the maximum distances (in feet) at which each
can read the sign. The data are provided below.
440
360
490
600
600
490
540
400
540
490
600
540
240
440
440
490
a. Verify the necessary conditions for computing a confidence interval for the population
mean distance. We are told that the sample was a random sample so we just need to check
if a normal model for the response ‘max distance’ for the population is reasonable.
Comments:
119
b. Compute the sample mean maximum distance and the standard error (without the outlier).
c. Use a 95% confidence interval to estimate the population mean maximum distance at which
all drivers can read the sign. Write a paragraph that interprets this interval and the
confidence level.
One-Sample Statistics
N
DISTANCE
15
Mean
497.3333
Std. Deviation
73.43283
Std. Error
Mean
18.96028
One-Sample Test
Tes t Value = 0
DISTANCE
t
26.230
df
14
Sig. (2-tailed)
.000
120
Mean
Difference
497.3333
95% Confidence
Interval of the
Difference
Lower
Upper
456.6676 537.9991
Here is the summary of the Confidence intervals for the big 5 parameters covered in
Chapters 10 and 11. We have now covered the one and two population proportion scenarios
and the one population mean scenario
Population Proportion
Two Population Proportions
Population Mean
Parameter
p1  p 2
Parameter
pˆ 1  pˆ 2
Statistic
Standard Error
Parameter

Statistic
x
Standard Error
p
p̂
Statistic
Standard Error
s.e.( pˆ ) 
pˆ (1  pˆ )
n
s.e.( pˆ 1  pˆ 2 ) 
Confidence Interval
s.e.( x ) 
Confidence Interval
x  t *s.e.( x )
df = n – 1
 pˆ 1  pˆ 2   z *s.e. pˆ 1  pˆ 2 
Conservative Conf. Interval
s
n
Confidence Interval
pˆ  z *s.e.( pˆ )
pˆ 
pˆ 1 (1  pˆ 1 ) pˆ 2 (1  pˆ 2 )

n1
n2
z*
Paired Confidence Interval
d  t *s.e.(d )
df = n – 1
2 n
 z* 

Sample Size n  

 2m 
2
Two Population Means
General
Parameter
Statistic
Standard Error
s.e.x1  x2  
1   2
x1  x 2
Pooled
Confidence Interval
x1  x2   t s.e.(x1  x2 )
x1  x 2
pooled s.e.x1  x2   s p
s12 s22

n1 n2
where s p 
*
1   2
Parameter
Statistic
Standard Error
1 1

n1 n2
(n1  1)s12  (n 2  1) s 22
n1  n 2  2
Confidence Interval
df = min( n1  1, n2  1)
x1  x2   t * pooled s.e.(x1  x2 )
121
df = n1  n2  2
Additional Notes
A place to … jot down questions you may have and ask during office hours, take a few extra notes, write
out an extra practice problem or summary completed in lecture, create your own short summary about
this chapter.
122