Chapter 7 Section 4(1).ppt

Section 7.4-1
Copyright © 2014, 2012, 2010 Pearson Education, Inc.
Lecture Slides
Elementary Statistics
Twelfth Edition
and the Triola Statistics Series
by Mario F. Triola

Section 7.4-2
Chapter 7
Estimates and Sample Sizes
7-1 Review and Preview
7-2 Estimating a Population Proportion
7-3 Estimating a Population Mean
7-4 Estimating a Population Standard Deviation or
Variance

Section 7.4-3
Key Concept
This section we introduce the chi-square
probability distribution so that we can construct
confidence interval estimates of a population
standard deviation or variance.
We also present a method for determining the
sample size required to estimate a population
standard deviation or variance.

Section 7.4-4
Chi-Square Distribution
In a normally distributed population with variance σ2 ,
assume that we randomly select independent samples of
size n and, for each sample, compute the sample variance
s2 (which is the square of the sample standard deviation s).
The sample statistic χ2 (pronounced chi-square) has a
sampling distribution called the chi-square distribution.

Section 7.4-5
where
n = sample size
s2
= sample variance
σ2 = population variance
df = n – 1 degrees of freedom
Chi-Square Distribution
  2
2
2
1
n s





Section 7.4-6
Properties of the Distribution
of the Chi-Square Statistic
1. The chi-square distribution is not symmetric, unlike
the normal and Student t distributions. As the number of
degrees of freedom increases, the distribution becomes
more symmetric.
Chi-Square Distribution Chi-Square Distribution for
df = 10 and df = 20

Section 7.4-7
2. The values of chi-square can be zero or positive, but it cannot
be negative.
3. The chi-square distribution is different for each number of
degrees of freedom, which is df = n – 1. As the number of
degrees of freedom increases, the chi-square distribution
approaches a normal distribution.
In Table A-4, each critical value of χ2 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.
Properties of the Distribution
of the Chi-Square Statistic

Section 7.4-8
Example
A simple random sample of 22 IQ scores is obtained.
Construction of a confidence interval for the population
standard deviation σ requires the left and right critical
values of χ2 corresponding to a confidence level of 95%
and a sample size of n = 22.
Find the critical χ2 values corresponding to a 95% level of
confidence.

Section 7.4-9
Example - Continued
Critical Values of the Chi-Square Distribution

Section 7.4-10
Estimators of σ2
The sample variance s2 is the best point
estimate of the population variance σ2.

Section 7.4-11
Estimator of σ
The sample standard deviation s is a
commonly used point estimate of σ (even
though it is a biased estimate).

Section 7.4-12
Confidence Interval for Estimating a
Population Standard Deviation or Variance
σ = population standard deviation
σ2 = population variance
s = sample standard deviation
s2 = sample variance
n = number of sample values (sample size)
E = margin of error
= left-tailed critical value of
= right-tailed critical value of
2
L

2
R

2

2


Section 7.4-13
Requirements:
1. The sample is a simple random sample.
2. The population must have normally distributed
values (even if the sample is large).

Section 7.4-14
Confidence Interval for the Population Variance σ2
n 1
 s2
R
2
 2

n 1
 s2
L
2

Section 7.4-15
Confidence Interval for the Population
Standard Deviation σ
n 1
 s2
R
2
  
n 1
 s2
L
2

Section 7.4-16
Procedure for Constructing a
Confidence Interval for σ or σ2
1. Verify that the required assumptions are satisfied.
2. Using n – 1 degrees of freedom, refer to Table A-4
or use technology to find the critical values
and that correspond to the desired
confidence level.
3. Evaluate the upper and lower confidence interval limits
using this format of the confidence interval:
n 1
 s2
R
2
 2

n 1
 s2
L
2
L
2
R
2

Section 7.4-17
4. If a confidence interval estimate of σ is desired, take the
square root of the upper and lower confidence interval
limits and change σ2 to σ.
5. Round the resulting confidence level limits. If using the
original set of data to construct a confidence interval,
round the confidence interval limits to one more decimal
place than is used for the original set of data. If using the
sample standard deviation or variance, round the
confidence interval limits to the same number of decimal
places.
Procedure for Constructing a
Confidence Interval for σ or σ2

Section 7.4-18
Caution
Confidence intervals can be used informally to compare
the variation in different data sets, but the overlapping of
confidence intervals should not be used for making
formal and final conclusions about equality of variances
or standard deviations.

Section 7.4-19
A group of 22 subjects took an IQ test during part of a study.
The subjects had a standard deviation IQ score of 14.3.
Construct a 95% confidence interval estimate of σ, the
standard deviation of the population from which the sample
was obtained.
Example

Section 7.4-20
We must first verify the requirements are met.
1. We can treat the sample as a simple random sample.
2. The following display shows a histogram of the data and the
normality assumption can be reasonably met.
Example - Continued

Section 7.4-21
n = 22 so df = 22 – 1 = 21
Use table A-4 to find:
Example - Continued
2 2
10 283 35 479
 
 
and
. .
L R
   
     
2 2
2
2 2
2 2
2
1 1
22 1 14 3 22 1 14 3
35 479 10 283

 

 
 
 
 
. .
. .
R L
n s n s

Section 7.4-22
Evaluation of the preceding expression yields:
Example - Continued
Finding the square root of each part (before rounding), then
rounding to two decimal places, yields this 95% confidence
interval estimate of the population standard deviation:
2
121 0 417 6

 
. .
11 0 20 4

 
. .
Based on this result, we have 95% confidence that the limits
of 11.0 and 20.4 contain the true value of σ.

Section 7.4-23
Determining Sample Sizes
The procedures for finding the sample size necessary to
estimate σ2 are much more complex than the procedures
given earlier for means and proportions. Instead of using
very complicated procedures, we will use Table 7-2.
STATDISK also provides sample sizes. With STATDISK,
select Analysis, Sample Size Determination, and then
Estimate St Dev.
Minitab, Excel, and the TI-83/84 Plus calculator do not
provide such sample sizes.

Section 7.4-24
Determining Sample Sizes

Section 7.4-25
We want to estimate the standard deviation σ of all voltage
levels in a home. We want to be 95% confident that our
estimate is within 20% of the true value of σ.
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 σ correspond to a sample of size 48.
We should obtain a simple random sample of 48 voltage
levels from the population of voltage levels.
Example

Chapter 7 Section 4(1).ppt

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