Transcript Chapter 7.3, 7.4, 7.5

Estimating a
Population
Mean: σ
Known
7-3, pg 355
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
1
Section 7.3: Estimation of a
population mean µ
(σ is known)
In this section we cover methods for
estimating a population mean. In addition
to knowing the values of the sample data or
statistics, we must also know the value of
the population standard deviation, .
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
2
Point Estimate of the Population
Mean
The sample mean x is the best point estimate
of the population mean µ.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
3
Confidence Interval for
Estimating a Population Mean
(with  Known)
 = population mean
 = population standard deviation
x = sample mean
n = number of sample values
E = margin of error
z/2 = z score separating an area of a/2 in the right
tail of the standard normal distribution
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
4
Requirements to check:
1. The value of the population standard
deviation  is known.
2. Either or both of these conditions is
satisfied: The population is normally
distributed or n > 30. (Just like in the
Central Limit Theorem.)
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
5
Confidence Interval for
Estimating a Population Mean
(with  Known)
x  E    x  E where E  z 2 
or
x E
or
x  E,x  E 
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.

n
6
Definition
The two values x – E and x + E are
called confidence interval limits.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
7
Round-Off Rule for Confidence
Intervals Used to Estimate µ
1. When using the original set of data, round the
confidence interval limits to one more decimal
place than used in original set of data.
2. When the original set of data is unknown and
only the summary statistics (n, x, s) are used,
round the confidence interval limits to the same
number of decimal places used for the sample
mean.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
8
Confidence Intervals by TI-83/84
•
•
•
•
•
•
•
•
•
Press STAT and select TESTS
Scroll down to ZInterval press ENTER
choose Data or Stats. For Stats:
Type in s: (known st. deviation)
_
x: (sample mean)
n: (sample size)
C-Level: (confidence level)
Press on Calculate
Read the confidence interval (…..,..…)
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
9
Confidence Intervals by Excel
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
10
Finding a Sample Size for Estimating
a Population Mean
σ = population standard deviation
E = desired margin of error
zα/2 = z score separating an area of /2 in the right tail of the
standard normal distribution
n=
(z/2)  
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
2
E
11
Round-Off Rule for Sample Size n
If the computed sample size n is
not a whole number, round the
value of n up to the next larger
whole number.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
12
Example Using Excel:
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
13
Estimating a
Population
Mean: σ Not
Known
7-4, pg 355
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
14
Section 7.4: Estimation of a
population mean µ
(σ is not known)
This section presents methods for
estimating a population mean when the
population standard deviation s is not
known.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
15
Sample Mean
_
The sample mean x is still the
best point estimate of the
population mean µ.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
16
Construction of a confidence
intervals for mean µ
(σ is not known)
With σ unknown, we use the
Student t distribution instead of
normal distribution.
It involves a new feature: number
of degrees of freedom
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
17
Definition
The number of degrees of freedom for a collection
of sample data is the number of sample values that
can vary after certain restrictions have been
imposed on all data values.
The degree of freedom is often abbreviated df.
degrees of freedom = n – 1
in this section.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
18
Margin of Error E for Estimate of µ
(With σ Not Known)
Formula 7-6
E = tα/2
s
n
where tα/2 has n – 1 degrees of freedom.
t/2 = critical t value separating an area of /2 in the
right tail of the t distribution
Table A-3 lists values for tα/2
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
19
Confidence Interval for the Estimate of
μ (With σ Not Known)
x–E <µ<x +E
where
E = tα/2 s
n
df = n – 1
tα/2 found in Table A-3
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
20
Important Properties of the
Student t Distribution
1. The Student t distribution is different for different sample sizes
(see the following slide, for the cases n = 3 and n = 12).
2. The Student t distribution has the same general symmetric bell shape
as the standard normal distribution but it reflects the greater
variability (with wider distributions) that is expected with small
samples
3. The Student t distribution has a mean of t = 0 (just as the standard
normal distribution has a mean of z = 0).
4. The standard deviation of the Student t distribution varies with the
sample size and is greater than 1 (unlike the standard normal
distribution, which has a σ = 1).
5. As the sample size n gets larger, the Student t distribution gets closer
to the normal distribution.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
21
Student t Distributions for
n = 3 and n = 12
Figure 7-5
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
22
Choosing the Appropriate Distribution
Use the normal (z)
distribution
Use t distribution
 known and normally
distributed population
or
 known and n > 30
 not known and
normally distributed
population
or
 not known and n > 30
Methods of Chapter 7 do Population is not normally
not apply
distributed and n ≤ 30
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
23
Confidence Intervals by TI-83/84
•
•
•
•
•
•
•
•
•
Press STAT and select TESTS
Scroll down to TInterval press ENTER
choose Data or Stats. For Stats:
_
Type in x: (sample mean)
Sx: (sample st. deviation)
n: (number of trials)
C-Level: (confidence level)
Press on Calculate
Read the confidence interval (…..,..…)
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
24
Confidence Intervals by TI-83/84
•
•
•
•
•
•
•
•
•
Press STAT and select TESTS
Scroll down to TInterval press ENTER
choose Data or Stats. For Data:
Type in List: L1 (or L2 or L3)
(specify the list containing your data)
Freq: 1 (leave it)
C-Level: (confidence level)
Press on Calculate
Read the confidence interval (…..,..…)
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
25
Confidence Intervals by Excel
Example 2, page 358, first calculate margin of error
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
26
Confidence Intervals by Excel
Example 2, page 358, first calculate margin of error
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
27
Finding the Point Estimate
and E from a Confidence Interval
Point estimate of µ:
x = (upper confidence limit) + (lower confidence limit)
2
Margin of Error:
E = (upper confidence limit) – (lower confidence limit)
2
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
28
Section 7-5
Estimating a Population
Variance
This section covers the estimation
2
of a population variance  and
standard deviation .
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
29
Estimator of 
2
The sample variance s2 is the best
point estimate of the population
variance 2.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
30
Estimator of 
The sample standard deviation s is a
commonly used point estimate of .
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
31
Construction of confidence
2
intervals for 
We use the chi-square distribution,
denoted by Greek character 2
(pronounced chi-square).
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
32
Properties of the Chi-Square
Distribution
1. The chi-square distribution is not symmetric, unlike
the normal and Student t distributions.
degrees of freedom = n – 1
Chi-Square Distribution
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
Chi-Square Distribution for
df = 10 and df = 20
33
Properties of the Chi-Square
Distribution
2. The values of chi-square can be zero or positive, but
they cannot be negative.
3. The chi-square distribution is different for each
number of degrees of freedom, which is df = n – 1.
In Table A-4, each critical value of 2 corresponds to
an area given in the top row of the table, and that
area represents the cumulative area located to the
right of the critical value.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
34
Example
A sample of ten voltage levels is obtained.
Construction of a confidence interval for the
population standard deviation  requires the
left and right critical values of 2
corresponding to a confidence level of 95%
and a sample size of n = 10.
Find the critical value of 2 separating an area
of 0.025 in the left tail, and find the critical
value of 2 separating an area of 0.025 in the
right tail.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
35
Example
Critical Values of the Chi-Square Distribution
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
36
Confidence Interval for Estimating a
Population Variance
n  1s

2
R
2

Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
2
n  1s



2
2
L
37
Confidence Interval for Estimating a
Population Standard Deviation
n  1s

2
R
2
 
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
n  1s

2
2
L
38
Requirement:
The population must have
normally distributed values
(even if the sample is large)
This requirement is very strict
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
39
Round-Off Rule for Confidence
Intervals Used to Estimate  or  2
1. When using the original set of data, round
the confidence interval limits to one more
decimal place than used in original set of
data.
2. When the original set of data is unknown
and only the summary statistics (n, x, s) are
used, round the confidence interval limits to
the same number of decimal places used for
the sample standard deviation.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
40
Determining Sample Sizes
The procedure for finding the sample size
necessary to estimate 2 is based on Table 7-2.
You just read the required sample size from an
appropriate line of the table.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
41
Determining Sample Sizes
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
42
Example:
We want to estimate the standard deviation .
We want to be 95% confident that our estimate
is within 20% of the true value of .
How large should the sample be?
Assume that the population is normally
distributed.
From Table 7-2, we can see that 95% confidence
and an error of 20% for  correspond to a
sample of size 48.
We should obtain a sample of 48 values.
Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved.
43