Transcript Ch. 6.3 PowerPt

ELEMENTARY
Section 6-3
STATISTICS
Estimating a Population Mean: Small Samples
EIGHTH
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION
MARIO F. TRIOLA
1
Small Samples
Assumptions
If 1) n  30
2) The sample is a simple random sample.
3) The sample is from a normally
distributed population.
Case 1 ( is known): Largely unrealistic; Use
methods from 6-2
Case 2 (is unknown): Use Student t
distribution
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
Student t Distribution
If the distribution of a population is
essentially normal, then the distribution of
t =
x-µ
s
n
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Student t Distribution
If the distribution of a population is
essentially normal, then the distribution of
t =
x-µ
s
n
 is essentially a Student t Distribution for all
samples of size n.
 is used to find critical values denoted by
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
t/ 2
4
Using the TI Calculator
Find the “t” score using the
InvT command as follows:
2nd - Vars – then
invT( percent, degree of
freedom )
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Definition
Degrees of Freedom (df )
corresponds to the number of
sample values that can vary after
certain restrictions have imposed
on all data values
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Definition
Degrees of Freedom (df )
corresponds to the number of sample values
that can vary after certain restrictions have
imposed on all data values
df = n - 1
in this section
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Definition
Degrees of Freedom (df ) = n - 1
corresponds to the number of sample values
that can vary after certain restrictions have
imposed on all data values
Any
Any
Any
Any
Any
#
#
#
#
#
n = 10
Any Any
#
#
Any
Any
Specific
#
#
#
df = 10 - 1 = 9
so that x = a specific number
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Margin of Error E for Estimate of 
Based on an Unknown  and a Small Simple Random
Sample from a Normally Distributed Population
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
9
Margin of Error E for Estimate of 
Based on an Unknown  and a Small Simple Random
Sample from a Normally Distributed Population
Formula 6-2
E = t /
s
2
n
where t/ 2 has n - 1 degrees of freedom
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Confidence Interval for the
Estimate of E
Based on an Unknown  and a Small Simple Random
Sample from a Normally Distributed Population
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Confidence Interval for the
Estimate of E
Based on an Unknown  and a Small Simple Random
Sample from a Normally Distributed Population
x-E <µ< x +E
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
Confidence Interval for the
Estimate of E
Based on an Unknown  and a Small Simple Random
Sample from a Normally Distributed Population
x-E <µ< x +E
where
E = t/2 s
n
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13
Confidence Interval for the
Estimate of E
Based on an Unknown  and a Small Simple Random
Sample from a Normally Distributed Population
x-E <µ< x +E
where
E = t/2 s
n
t/2 found using invT
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
14
Properties of the Student t Distribution

There is a different distribution for different
sample sizes

It has the same general bell shape as the normal
curve but greater variability because of small
samples.

The t distribution has a mean of t = 0

The standard deviation of the t distribution
varies with the sample size and is greater than 1

As the sample size (n) gets larger, the t
distribution gets closer to the normal
distribution.
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
15
Student t Distributions for
n = 3 and n = 12
Student t
Standard
normal
distribution
distribution
with n = 12
Student t
distribution
with n = 3
Figure 6-5
0
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
16
Using the Normal and t Distribution
Figure 6-6
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
17
Example:
A study of 12 Dodge Vipers involved in
collisions resulted in repairs averaging $26,227 and a
standard deviation of $15,873. Find the 95% interval
estimate of , the mean repair cost for all Dodge Vipers
involved in collisions. (The 12 cars’ distribution appears to
be bell-shaped.)
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
18
Example:
A study of 12 Dodge Vipers involved in
collisions resulted in repairs averaging $26,227 and a
standard deviation of $15,873. Find the 95% interval
estimate of , the mean repair cost for all Dodge Vipers
involved in collisions. (The 12 cars’ distribution appears to
be bell-shaped.)
x = 26,227
s = 15,873
 = 0.05
/2 = 0.025
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
19
Example:
A study of 12 Dodge Vipers involved in
collisions resulted in repairs averaging $26,227 and a
standard deviation of $15,873. Find the 95% interval
estimate of , the mean repair cost for all Dodge Vipers
involved in collisions. (The 12 cars’ distribution appears to
be bell-shaped.)
x = 26,227
s = 15,873
E = t/2 s = (2.201)(15,873) = 10,085.29
n
12
 = 0.05
/2 = 0.025
t/2 = 2.201
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
20
Example:
A study of 12 Dodge Vipers involved in
collisions resulted in repairs averaging $26,227 and a
standard deviation of $15,873. Find the 95% interval
estimate of , the mean repair cost for all Dodge Vipers
involved in collisions. (The 12 cars’ distribution appears to
be bell-shaped.)
x = 26,227
s = 15,873
 = 0.05
/2 = 0.025
t/2 = 2.201
E = t/2 s = (2.201)(15,873) = 10,085.3
12
n
x -E
<µ<
x +E
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
21
Example:
A study of 12 Dodge Vipers involved in
collisions resulted in repairs averaging $26,227 and a
standard deviation of $15,873. Find the 95% interval
estimate of , the mean repair cost for all Dodge Vipers
involved in collisions. (The 12 cars’ distribution appears to
be bell-shaped.)
x = 26,227
s = 15,873
 = 0.05
/2 = 0.025
t/2 = 2.201
E = t/2 s = (2.201)(15,873) = 10,085.3
n
x -E
<µ<
26,227 - 10,085.3 < µ <
12
x +E
26,227 + 10,085.3
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
22
Example:
A study of 12 Dodge Vipers involved in
collisions resulted in repairs averaging $26,227 and a
standard deviation of $15,873. Find the 95% interval
estimate of , the mean repair cost for all Dodge Vipers
involved in collisions. (The 12 cars’ distribution appears to
be bell-shaped.)
x = 26,227
s = 15,873
 = 0.05
/2 = 0.025
t/2 = 2.201
E = t/2 s = (2.201)(15,873) = 10,085.3
n
x -E
<µ<
26,227 - 10,085.3 < µ <
$16,141.7 < µ <
12
x +E
26,227 + 10,085.3
$36,312.3
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
23
Example:
A study of 12 Dodge Vipers involved in
collisions resulted in repairs averaging $26,227 and a
standard deviation of $15,873. Find the 95% interval
estimate of , the mean repair cost for all Dodge Vipers
involved in collisions. (The 12 cars’ distribution appears to
be bell-shaped.)
x = 26,227
s = 15,873
 = 0.05
/2 = 0.025
t/2 = 2.201
E = t/2 s = (2.201)(15,873) = 10,085.3
n
x -E
<µ<
26,227 - 10,085.3 < µ <
$16,141.7 < µ <
12
x +E
26,227 + 10,085.3
$36,312.3
We are 95% confident that this interval contains the
average cost of repairing a Dodge Viper.
Chapter 6. Section 6-3. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
24