Transcript 07-w11-stats250-bgunderson-chapter-11-ci-for-a

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 [email protected] 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.
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.
What parameter are we learning how to estimate
and test hypotheses about now?
(CLICKER in your answer)
A)
m
B)
x
Recall our results about Sample Means
page 109
Sample Means
Mean: EX    X  
Standard Deviation: s.d .( X )   X 

n
Sampling Distribution of X

 

.
N

,
If X has the N (  ,  ) distribution, then X is 

n

If X follows any distribution with mean  and standard deviation  and n is large,

 

.
N

,
then X is approximately 

n

This last result is the Central Limit Theorem
More on the Standard Deviation of x
Standard deviation of the sample mean:
s.d .( x )  
n
Interpret: approximately the average distance of the
possible x values (for repeated samples of same size n)
from the true population mean .
Note: sample size increases  standard deviation decreases.
Problem: in practice we would not know the value of .
The Standard Error of
x
Standard error of the sample mean:
s.e.( x ) 
s
n
Interpret: Estimates, approximately, the average distance
of the possible x values (for repeated samples of same
size n) from the true population mean .
9.9 Standardized Statistics for Means

pg 112
z-statistic for a sample mean:
z=
x

n
Dilemma =
has (approximately) a standard
normal distribution N(0,1).
9.9 Standardized Statistics for Means

Replace  with s …
x
s
n
won’t be approximately N(0,1)
instead it has a ….
Student’s t-Distribution
About t-distributions ...





Symmetric, unimodal, centered at 0
Flatter with heavier tails compared to the N(0,1)
As df increases … t distribution  N(0,1)
Can still use ideas about standard scores.
Tables A.2 & A.3
summarize percentiles
for t-distributions
9.10 Sampling Distribution for any Statistic
page 113

The sampling distribution of a statistic … is the
distribution of possible values of the statistic for
repeated samples of same size from a population.

Every statistic has a sampling distribution, but
the appropriate distribution may not always be
normal, or even be approximately bell-shaped.

Example 9.19 (pg 377-378) shows good example.
11.1 Intro to CIs for Means
page 115

Interval = a range of reasonable values for the
parameter with an associated high level of
confidence. “We are 95% confident that …”

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: CI for a Population Mean 





Design of a highway sign. Q: What is the mean maximum
distance at which drivers are able to read the sign?
Data: Researcher will take a r.s. of n = 16 drivers and measure
the maximum distances (in feet) at which each can read the sign.
Population parameter: m = _______________ 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.
Recall Sampling Distribution for Mean
pg 116
If x is sample mean for a random sample of size n from a
normal model, then the distribution of the sample mean is:
Central Limit Theorem
If x is the sample mean from a random sample of size n
from a population with any model, with mean m,
and standard deviation s, then when n is large, the
sampling distribution of the sample mean is approximately
.
The Standard Error of the Sample Mean
s.e.( x ) 
Estimates, roughly, the average distance of the possible
values from the true population proportion .
One-sample t Confidence Interval for 
x  t s.e.( x )
*
where t* is from a t(n – 1) distribution.
Interval requires have a r.s. from normal popul.
If sample size large (n > 30), assumption of
normality not so crucial and result is approx.
x
Try It! Using Table A.2
page 117
(a) Find t* for a 90% CI
based on n = 12 obs.
(b) Find t* for a 95% CI
based on n = 30 obs.
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition. 2012.
Used with permission.
Try It! Using Table A.2
(c) Find t* for a 95% CI
based on n = 54 obs.
(d) What happens to t*
as n gets larger?
From Utts, Jessica M. and Robert F. Heckard. Mind on Statistics, Fourth Edition.
2012.
Used with permission.
Try It! CI for Mean Maximum Distance pg 119

r.s. of n = 16 drivers; measured max distance
(in ft) at which can read sign.
440 490 600 540 540 600 240 440
360 600 490 400 490 540 440 490
a. Verify conditions. Told sample was a random
sample so need to check if normal model for
‘max distance’ for the population is
reasonable.
5
700
Maximum Distance
Count
CI for Mean Maximum Distance: Check conditions
4
650
600
550
3
500
450
2
400
350
1
300
0
250
225
275
325
375
425
475
525
575
625
200
Maximum Distance (in feet)
Normal Q-Q Plot of DISTANCE
700
600
500
400
300
200
300
Observed Value
400
500
600
700
CI for Mean Maximum Distance: Std Error
r.s. of n = 15 drivers; max distance (in ft)
440 490 600 540 540 600 240 440
360 600 490 400 490 540 440 490
b. Compute sample mean max distance and std error.

CI for Mean Maximum Distance: Interpret
c. Use 95% CI to estim the mean maximum distance at
which all drivers can read the sign. Write a paragraph
that interprets this interval and the confidence level.
CI for Mean Maximum Distance: SPSS output
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
Mean
Difference
497.3333
95% Confidence
Interval of the
Difference
Lower
Upper
456.6676 537.9991
Q3: Environmental Proposal – Consider the following table that
summarizes the attitude (in favor, indifferent, or opposed) to a
particular environmental proposal and the political party for the
100 U.S. senators.
In Favor Indifferent Opposed
Democrat
Republican
Total
27
13
40
15
10
25
18
17
35
Total
60
40
100
a. What is the probability that a randomly selected
Democrat will be in favor of the proposal?
In Favor Indifferent Opposed
Q3: Environmental
Proposal
Democrat
Republican
Total
27
13
40
15
10
25
18
17
35
Total
60
40
100
a. What is the probability that a randomly selected Democrat
will be in favor of the proposal?
b. To determine whether being a Democrat was independent
from being in favor, you would compare answer to part (a)
with another probability. What is it?
Hence, being a Democrat and being in favor are:
dependent
independent
Q4: Satisfied with your first-year experience? – Ugrads today are
less satisfied with first year college experience than 5 years ago.
Survey based on sample of 500 undergraduates completing first
year. Fifty-eight percent reported they were happy, down from
63 percent (established rate 5 years prior). Is result sufficient
evidence to conclude ‘satisfied’ rate is significantly lower than
the prior rate of 63%? Use a 5% significance level
a. Following defin of parameter of interest is not complete,
so you are asked to correct it.
Let p = the population proportion
of all first year college students
Q4: Satisfied with your first-year experience? – Is result sufficient
evidence to conclude ‘satisfied’ rate is significantly lower than the
prior rate of 63%? Use a 5% significance level.
b. State the appropriate hypotheses.
H0: ________________________
Ha:_________________________
Q4: Satisfied with your first-year experience? – Is result sufficient
evidence to conclude ‘satisfied’ rate is significantly lower than the
prior rate of 63%? Use a 5% significance level.
c. Test statistic value was z = -2.32. Find p-value.
There (circle one)
is
is not
sufficient evidence to conclude that the
satisfied rate is significantly lower than prior rate of 63%.