M ARIO F . T RIOLA
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1Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH
EDITIONEDITION
ELEMENTARY STATISTICSChapter 6 Estimates and Sample Sizes
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2Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Chapter 6Estimates and Sample Sizes
6-1 Overview
6-2 Estimating a Population Mean: Large Samples
6-3 Estimating a Population Mean: Small Samples
6-4 Sample Size Required to Estimate µ
6-5 Estimating a Population Proportion
6-6 Estimating a Population Variance
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3Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6-1 Overview
methods for estimating population means, proportions, and variances
methods for determining sample sizes
This chapter presents:
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4Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6-2
Estimating a Population Mean:Large Samples
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5Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Assumptions
n > 30The sample must have more than 30 values.
Simple Random SampleAll samples of the same size have an equal chance
of being selected.
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6Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Assumptions
n > 30The sample must have more than 30 values.
Simple Random SampleAll samples of the same size have an equal chance
of being selected.
Data collected carelessly can be absolutely worthless, even if the sample
is quite large.
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7Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Definitions Estimator
a formula or process for using sample data to estimate a population parameter
Estimatea specific value or range of values used to approximate some population parameter
Point Estimatea single value (or point) used to approximate a population parameter
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8Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Estimatora formula or process for using sample data to estimate a population
parameter
Estimatea specific value or range of values used to approximate
some population parameter
Point Estimatea single value (or point) used to approximate a population
parameter
The sample mean x is the best point estimate of the population mean µ.
Definitions
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9Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
DefinitionConfidence Interval
(or Interval Estimate)
a range (or an interval) of values used to estimate the true value of the population
parameter
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10Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
DefinitionConfidence Interval
(or Interval Estimate)
a range (or an interval) of values used to estimate the true value of the population
parameter
Lower # < population parameter < Upper #
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11Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
DefinitionConfidence Interval
(or Interval Estimate)
a range (or an interval) of values used to estimate the true value of the population
parameter
Lower # < population parameter < Upper #
As an exampleLower # < < Upper #
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12Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
the probability 1 - (often expressed as the equivalent percentage value) that is the relative frequency of times the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times
usually 90%, 95%, or 99% ( = 10%), ( = 5%), ( = 1%)
DefinitionDegree of Confidence
(level of confidence or confidence coefficient)
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13Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Interpreting a Confidence Interval
Correct: We are 95% confident that the interval from 98.08 to 98.32 actually does contain the true value of
. This means that if we were to select many different samples of size 106 and construct the confidence intervals, 95% of them would actually contain the
value of the population mean .
Wrong: There is a 95% chance that the true value of will fall between 98.08 and 98.32.
98.08o < µ < 98.32o
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14Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Figure 6-1
Confidence Intervals from 20 Different Samples
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15Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
the 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.
Definition
Critical Value
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16Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
The Critical Value
z=0
Figure 6-2Found from Table A-2
(corresponds to area of
0.5 - 2 )
z2
z2-z2
2 2
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17Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
-z2z2
95%
.95
.025.025
2 = 2.5% = .025 = 5%
Finding z2 for 95% Degree of Confidence
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18Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
-z2z2
95%
.95
.025.025
2 = 2.5% = .025 = 5%
Critical Values
Finding z2 for 95% Degree of Confidence
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19Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
.4750
.025
Use Table A-2 to find a z score of 1.96
= 0.025 = 0.05
Finding z2 for 95% Degree of Confidence
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20Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Finding z2 for 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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21Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Margin of Error is the maximum likely difference observed between sample mean x and true population
mean µ.
denoted by E
Definition
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22Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Margin of Error is the maximum likely difference observed between sample mean x and true population
mean µ.
denoted by E
µ x + Ex - E
Definition
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23Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Margin of Error is the maximum likely difference observed between sample mean x and true population
mean µ.
denoted by E
µ x + Ex - E
x -E < µ < x +E
Definition
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24Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Margin of Error is the maximum likely difference observed between sample mean x and true population
mean µ.
denoted by E
µ x + Ex - E
x -E < µ < x +Elower limit
Definition
upper limit
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25Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Definition Margin of Error
µ x + Ex - E
E = z/2 • Formula 6-1n
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26Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Definition Margin of Error
µ x + Ex - E
also called the maximum error of the estimate
E = z/2 • Formula 6-1n
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27Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Calculating E When Is Unknown
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.
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28Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Confidence Interval (or Interval Estimate)
for Population Mean µ(Based on Large Samples: n >30)
x - E < µ < x + E
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29Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
x - E < µ < x + E
µ = x + E
Confidence Interval (or Interval Estimate)
for Population Mean µ(Based on Large Samples: n >30)
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30Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
x - E < µ < x + E
µ = x + E
(x + E, x - E)
Confidence Interval (or Interval Estimate)
for Population Mean µ(Based on Large Samples: n >30)
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31Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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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32Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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 the population standard deviation is unknown, use the value of the sample standard deviation s provided that n > 30.
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33Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
3. Find the values of x - E and x + E. Substitute thosevalues in the general format of the confidenceinterval: x - E < µ < x + E
2. Evaluate the margin of error E = z2 • / n . If the population standard deviation is unknown, use the value of the sample standard deviation s provided that n > 30.
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34Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
3. Find the values of x - E and x + E. Substitute thosevalues in the general format of the confidenceinterval:
4. Round using the confidence intervals roundoff rules.
x - E < µ < x + E
2. Evaluate the margin of error E = z2 • / n . If the population standard deviation is unknown, use the value of the sample standard deviation s provided that n > 30.
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35Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Round-Off Rule for Confidence
Intervals Used to Estimate µ1. When using the original set of data, round the confidence interval limits to one more decimal place than used in original set of data.
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36Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
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.
Round-Off Rule for Confidence
Intervals Used to Estimate µ
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37Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.
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38Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n = 106
x = 98.2o
s = 0.62o
= 0.05/2 = 0.025
z / 2 = 1.96
Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.
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39Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n = 106
x = 98.20o
s = 0.62o
= 0.05/2 = 0.025
z / 2 = 1.96
E = z / 2 • = 1.96 • 0.62 = 0.12n 106
Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.
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40Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n = 106
x = 98.20o
s = 0.62o
= 0.05/2 = 0.025
z / 2 = 1.96
E = z / 2 • = 1.96 • 0.62 = 0.12n 106
x - E < < x + E
Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.
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41Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n = 106
x = 98.20o
s = 0.62o
= 0.05/2 = 0.025
z / 2 = 1.96
E = z / 2 • = 1.96 • 0.62 = 0.12n 106
x - E < < x + E98.20o - 0.12 < < 98.20o + 0.12
Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.
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42Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n = 106
x = 98.20o
s = 0.62o
= 0.05/2 = 0.025
z / 2 = 1.96
E = z / 2 • = 1.96 • 0.62 = 0.12n 106
x - E < < x + E98.20o - 0.12 < < 98.20o + 0.12
98.08o < < 98.32o
Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.
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43Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
n = 106
x = 98.20o
s = 0.62o
= 0.05/2 = 0.025
z / 2 = 1.96
E = z / 2 • = 1.96 • 0.62 = 0.12n 106
x - E < < x + E98.08o < < 98.32o
Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.
Based on the sample provided, the confidence interval for the
population mean is 98.08o < < 98.32o. If we were to select many different samples of the same size, 95% of the confidence intervals
would actually contain the population mean .
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44Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Finding the Point Estimate and E from a Confidence Interval
Point estimate of µ:
x = (upper confidence interval limit) + (lower confidence interval limit)
2
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45Chapter 6. Section 6-1 and 6-2. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Finding the Point Estimate and E from a Confidence Interval
Point estimate of µ:
x = (upper confidence interval limit) + (lower confidence interval limit)
2
Margin of Error:
E = (upper confidence interval limit) - (lower confidence interval limit)
2