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![Page 1: Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Estimates and Sample Sizes Chapter 6 M A R I O F. T R I O L A Copyright © 1998,](https://reader035.fdocuments.in/reader035/viewer/2022081520/5697bf791a28abf838c82590/html5/thumbnails/1.jpg)
Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 1
Estimates and Sample Estimates and Sample SizesSizesChapter 6Chapter 6
M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 2
Chapter 6Estimate and Sample Sizes
6-1 Overview
6-2 Estimating a Population Mean: Large Samples
6-3 Estimating a Population Mean: Small Samples
6-4 Determining Sample Size
6-5 Estimating a Population Proportion
6-6 Estimating a Population Variance
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 3
6-1 Overview
This chapter presents:methods for estimating population means,
proportions, and variances
methods for determining sample sizes necessary to estimate the above parameters.
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 4
6-2 Estimating a Population Mean: Large Samples
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 5
Definitions Estimator
a sample statistic used to approximate a population parameter
Estimatea specific value or range of values used to approximate some population parameter
Point Estimatea single volume (or point) used to approximate a population parameter
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 6
Definitions Estimator
a sample statistic used to approximate a population parameter
Estimatea specific value or range of values used to approximate some
population parameter
Point Estimatea single volue (or point) used to approximate a popular
parameter
The sample mean x
population mean µ.is the best point estimate of the
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 7
Confidence Interval (or Interval Estimate)
a range (or an interval) of values likely to contain the true value of the population
parameter
Lower # < population parameter < Upper #
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 8
Confidence Interval (or Interval Estimate)
a range (or an interval) of values likely to contain the true value of the population
parameter
Lower # < population parameter < Upper #
As an exampleLower # < µ < Upper #
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 9
Definition Degree of Confidence
(level of confidence or confidence coefficient)
the probability 1 – (often expressed as the equivalent percentage value) that the confidence interval contains the true value of the population parameter
usually 95% or 99% ( = 5%) ( = 1%)
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 10
Confidence Intervals from Different Samples
98.00 98.50
• •x
98.08 98.32
• •• •
This confidence intervaldoes not contain µ
• •• •
µ = 98.25 (but unknown to us)
Figure 6-3
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 11
Definition Critical Valuethe number on the borderline separating sample
statistics that are likely to occur from those that
are unlikely to occur. The number z/2 is a critical
value that is a z score with the property that it
separates an area /2 in the right tail of the standard normal distribution. There is an area of
1 – between the vertical borderlines at
–z/2 and z/2.
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 12
If Degree of Confidence = 95%
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 13
If Degree of Confidence = 95%
–z2z2
95%
.95
.025.025
2 = 2.5% = .025 = 5%
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 14
If Degree of Confidence = 95%
–z2z2
95%
.95
.025.025
2 = 2.5% = .025 = 5%
Critical Values
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 15
95% Degree of Confidence
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 16
95% Degree of Confidence
.4750
.025
Use Table A-2 to find a z score of 1.96
= 0.025 = 0.05
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 17
95% Degree of Confidence
.025.025
–1.96 1.96
z2 = 1.96
.4750
.025
Use Table A-2 to find a z score of 1.96
= 0.025 = 0.05
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 18
Definition Margin of Error
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 19
Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true
population mean µ.
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 20
Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true
population mean µ.
denoted by E
µ x + Ex – E
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 21
x – E
Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true
population mean µ.
denoted by E
µ x + E
x – E < µ < x + E
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 22
Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true
population mean µ.
denoted by E
µ x + Ex – E
x – E < µ < x + E
lower limit
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 23
Definition Margin of Error is the maximum likely difference between the observed sample mean, x, and true
population mean µ.
denoted by E
µ x + Ex – E
x – E < µ < x +E
upper limitlower limit
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 24
Calculating the Margin of Error When Is Unknown
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 25
Calculating the Margin of Error When Is known
If n > 30, we can replace in Formula 6-1 by the sample standard deviation s.
If n 30, the population must have a normal distribution and we must know to use Formula 6-1.
E = z/2 • Formula 6-1n
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 26
Confidence Interval
• 1. If using original data, round to one more decimal place than used in data.
Round off Rules
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 27
Confidence Interval
• 1. If using original data, round to one more decimal place than used in data.
• 2. If given summary statistics (n, x, s), round to same number of decimal places
as in x.
Round off Rules
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 28
Procedure for Constructing a Confidence Interval for µ
( based on a large sample: n > 30 )
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 29
Procedure for Constructing a Confidence Interval for µ
( based on a large sample: n > 30 )
1. Find the critical value z2 that corresponds to the desired degree of confidence.
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 30
Procedure for Constructing a Confidence Interval for µ
( based on a large sample: n > 30 )
1. Find the critical value z2 that corresponds to the desired degree of confidence.
2. Evaluate the margin of error E= z2 • n . If thepopulation standard deviation is unknown andn > 30, use the value of the sample standarddeviation s.
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 31
Procedure for Constructing a Confidence Interval for µ
( based on a large sample: n > 30 )
1. Find the critical value z2 that corresponds to the desired degree of confidence.
2. Evaluate the margin of error E = z2 • n . If thepopulation standard deviation is unknown andn > 30, use the value of the sample standarddeviation s.
3. Find the values of x – E and x + E. Substitute thosevalues in the general format of the confidenceinterval:
x – E < µ < x + E
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 32
Procedure for Constructing a Confidence Interval for µ
( based on a large sample: n > 30 )
1. Find the critical value z2 that corresponds to the desired degree of confidence.
2. Evaluate the margin of error E= z2 • n . If thepopulation standard deviation is unknown andn > 30, use the value of the sample standarddeviation s.
3. Find the values of x – E and x + E. Substitute thosevalues in the general format of the confidenceinterval:
x – E < µ < x + E 4. Round using the confidence intervals round off rules.
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 33
Confidence Intervals from Different Samples
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 34
6-3
Determining Sample Size
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 35
Determining Sample Size
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 36
Determining Sample Size
z/ 2 •E =
n
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 37
Determining Sample Size
z/ 2 •E =
n
(solve for n by algebra)
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 38
Determining Sample Size
If n is not a whole number, round it up to the next higher whole number.
z/ 2 •E =
n
(solve for n by algebra)
z/ 2 E
n =
2Formula 6-2
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 39
What happens when E is doubled ?
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 40
What happens when E is doubled ?
/ 2 z
1
n = =2
1/ 2
(z ) 2
E = 1 :
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 41
What happens when E is doubled ?
Sample size n is decreased to 1/4 of its original value if E is doubled.
Larger errors allow smaller samples.
Smaller errors require larger samples.
/ 2 z
1
n = =2
1/ 2
(z ) 2
/ 2 z
2
n = =
2
4/ 2
(z ) 2
E = 1 :
E = 2 :
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Copyright © 1998, Triola, Elementary Statistics
Addison Wesley Longman 42
What if is unknown ?
1. Use the range rule of thumb to estimate the standard deviation as follows: range / 4
or
2. Calculate the sample standard deviation s and use it in place of . That value can be refined as more sample data are obtained.