slides by john loucks st . edward’s university
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Slides by John Loucks St . Edward’s University. Chapter 8 Interval Estimation. Population Mean: s Known. Population Mean: s Unknown. Determining the Sample Size. Population Proportion. Margin of Error and the Interval Estimate. - PowerPoint PPT PresentationTRANSCRIPT
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Slides by
JohnLoucks
St. Edward’sUniversity
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Chapter 8Interval Estimation
Population Mean: s Known Population Mean: s Unknown Determining the Sample Size Population Proportion
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A point estimator cannot be expected to provide the exact value of the population parameter.
An interval estimate can be computed by adding and subtracting a margin of error to the point estimate.
Point Estimate +/- Margin of Error
The purpose of an interval estimate is to provide information about how close the point estimate is to the value of the parameter.
Margin of Error and the Interval Estimate
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The general form of an interval estimate of a population mean is
Margin of Errorx
Margin of Error and the Interval Estimate
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Interval Estimate of a Population Mean:s Known
In order to develop an interval estimate of a population mean, the margin of error must be computed using either:• the population standard deviation s , or• the sample standard deviation s
s is rarely known exactly, but often a good estimate can be obtained based on historical data or other information.
We refer to such cases as the s known case.
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There is a 1 - probability that the value of asample mean will provide a margin of error of or less.
z x /2
/2/2 /2/21 - of all values1 - of all valuesx
Sampling distribution of
Sampling distribution of x
x
z x /2z x /2
Interval Estimate of a Population Mean:s Known
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/2/2 /2/21 - of all values1 - of all valuesx
Sampling distribution of
Sampling distribution of x
x
z x /2z x /2
[------------------------- -------------------------]
[------------------------- -------------------------]
[------------------------- -------------------------]
xx
x
intervaldoes not
include m
intervalincludes
m
intervalincludes
m
Interval Estimate of a Population Mean:s Known
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Interval Estimate of m
Interval Estimate of a Population Mean:s Known
x zn
/2
where: is the sample mean 1 - is the confidence coefficient z/2 is the z value providing an area of /2 in the upper tail of the standard
normal probability distribution s is the population standard deviation n is the sample size
x
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Interval Estimate of a Population Mean:s Known
Values of za/2 for the Most Commonly Used Confidence Levels
90% .10 .05 .9500 1.645
Confidence Table Level a a/2 Look-up Area za/2
95% .05 .025 .9750 1.960 99% .01 .005 .9950 2.576
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Meaning of Confidence
Because 90% of all the intervals constructed using will contain the population mean, we say we are 90% confident that the interval includes the population mean m.
xx 645.1
xx 645.1
We say that this interval has been established at the 90% confidence level.
The value .90 is referred to as the confidence coefficient.
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Interval Estimate of a Population Mean: Known
Example: Discount Sounds Discount Sounds has 260 retail outlets
throughoutthe United States. The firm is evaluating a
potentiallocation for a new outlet, based in part, on the
meanannual income of the individuals in the
marketingarea of the new location.
A sample of size n = 36 was taken; the sample
mean income is $41,100. The population is not
believed to be highly skewed. The population standard deviation is estimated to be $4,500,
and theconfidence coefficient to be used in the
interval estimate is .95.
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95% of the sample means that can be observedare within + 1.96 of the population mean . xThe margin of error is:
/ 2
4,5001.96 1,470
36z
n
Thus, at 95% confidence, the margin of error is $1,470.
Interval Estimate of a Population Mean: Known
Example: Discount Sounds
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Interval estimate of is:
Interval Estimate of a Population Mean: Known
We are 95% confident that the interval contains thepopulation mean.
$41,100 + $1,470or
$39,630 to $42,570
Example: Discount Sounds
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Using Excel to Construct aConfidence Interval: s Known Case
Excel Formula WorksheetA B C
1 Income Sample Size =COUNT(A2:A37)2 45,600 Sample Mean =AVERAGE(A2:A37)3 39,6014 40,035 Population Std. Dev. 45005 41,735 Confidence Coeff. 0.956 37,600 Lev. of Significance =1-C57 46,0808 48,925 Margin of Error =CONFIDENCE.NORM(C6,C4,C1)9 45,350
10 35,900 Point Estimate =C211 37,550 Lower Limit =C10-C812 34,745 Upper Limit =C10+C8
Note:Rows 13-37
are not shown.
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Using Excel to Construct aConfidence Interval: s Known Case
Excel Value WorksheetA B C
1 Income Sample Size 362 45,600 Sample Mean 41,100.003 39,6014 40,035 Population Std. Dev. 45005 41,735 Confidence Coeff. 0.956 37,600 Lev. of Significance 0.957 46,0808 48,925 Margin of Error 1,470.009 45,350
10 35,900 Point Estimate 41,100.0011 37,550 Lower Limit 39,630.0012 34,745 Upper Limit 42,570.00
Note:Rows 13-37
are not shown.
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Interval Estimate of a Population Mean: Known
90% 3.29 78.71 to 85.29
Confidence Margin Level of Error Interval Estimate
95% 3.92 78.08 to 85.92 99% 5.15 76.85 to 87.15
Example: Discount Sounds
In order to have a higher degree of confidence,the margin of error and thus the width of theconfidence interval must be larger.
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Interval Estimate of a Population Mean:s Known
Adequate Sample Size
In most applications, a sample size of n = 30 is adequate.
If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended.
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Interval Estimate of a Population Mean:s Known
Adequate Sample Size (continued)
If the population is believed to be at least approximately normal, a sample size of less than 15 can be used.
If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 will suffice.
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Interval Estimate of a Population Mean:s Unknown
If an estimate of the population standard deviation s cannot be developed prior to sampling, we use the sample standard deviation s to estimate s . This is the s unknown case.
In this case, the interval estimate for m is based on the t distribution.
(We’ll assume for now that the population is normally distributed.)
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William Gosset, writing under the name “Student”, is the founder of the t distribution.
t Distribution
Gosset was an Oxford graduate in mathematics and worked for the Guinness Brewery in Dublin.
He developed the t distribution while working on small-scale materials and temperature experiments.
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The t distribution is a family of similar probability distributions.
t Distribution
A specific t distribution depends on a parameter known as the degrees of freedom.
Degrees of freedom refer to the number of independent pieces of information that go into the computation of s.
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t Distribution
A t distribution with more degrees of freedom has less dispersion.
As the degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller.
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t Distribution
Standardnormal
distribution
t distribution(20 degreesof freedom)
t distribution(10 degrees
of freedom)
0
z, t
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For more than 100 degrees of freedom, the standard normal z value provides a good approximation to the t value.
t Distribution
The standard normal z values can be found in the infinite degrees ( ) row of the t distribution table.
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t Distribution
Degrees Area in Upper Tail
of Freedom .20 .10 .05 .025 .01 .005
. . . . . . .
50 .849 1.299 1.676 2.009 2.403 2.678
60 .848 1.296 1.671 2.000 2.390 2.660
80 .846 1.292 1.664 1.990 2.374 2.639
100 .845 1.290 1.660 1.984 2.364 2.626
.842 1.282 1.645 1.960 2.326 2.576Standard normalz values
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Interval Estimate
x tsn
/2
where: 1 - = the confidence coefficient t/2 = the t value providing an area of /2
in the upper tail of a t distribution with n - 1 degrees of freedom s = the sample standard deviation
Interval Estimate of a Population Mean:s Unknown
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A reporter for a student newspaper is writing an
article on the cost of off-campus housing. A sample
of 16 efficiency apartments within a half-mile of
campus resulted in a sample mean of $750 per month
and a sample standard deviation of $55.
Interval Estimate of a Population Mean:s Unknown
Example: Apartment Rents
Let us provide a 95% confidence interval estimate
of the mean rent per month for the population of
efficiency apartments within a half-mile of campus.
We will assume this population to be normallydistributed.
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At 95% confidence, = .05, and /2 = .025.
In the t distribution table we see that t.025 = 2.131.
t.025 is based on n - 1 = 16 - 1 = 15 degrees of freedom.
Interval Estimate of a Population Mean:s Unknown
Degrees Area in Upper Tail
of Freedom .20 .100 .050 .025 .010 .005
15 .866 1.341 1.753 2.131 2.602 2.947
16 .865 1.337 1.746 2.120 2.583 2.921
17 .863 1.333 1.740 2.110 2.567 2.898
18 .862 1.330 1.734 2.101 2.520 2.878
19 .861 1.328 1.729 2.093 2.539 2.861
. . . . . . .
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x tsn
.025
We are 95% confident that the mean rent per monthfor the population of efficiency apartments within ahalf-mile of campus is between $720.70 and $779.30.
Interval Estimate
Interval Estimate of a Population Mean:s Unknown
55750 2.131 650 29.30
16
Marginof Error
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Step 1 Click the Data tab on the Ribbon
Step 2 In the Analysis group, click Data Analysis
Step 3 Choose Descriptive Statistics from the list of Analysis Tools
Step 4 When the Descriptive Statistics dialog box appears: (see details on next slide)
Using Excel’sDescriptive Statistics Tool
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Using Excel’sDescriptive Statistics Tool
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Using Excel’sDescriptive Statistics Tool
Excel Value Worksheet
PointEstimat
e
Marginof Error
A B C D1 Rent Rent2 7753 810 Mean 6504 720 Standard Error 13.753485 700 Median 6646 670 Mode #N/A7 795 Standard Deviation 55.013948 768 Sample Variance 3026.5339 660 Kurtosis -0.62925
10 750 Skewness -0.7727411 645 Range 17012 805 Minimum 54513 780 Maximum 71514 790 Sum 1040015 815 Count 1616 757 Confidence Level (95%) 29.31488
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Interval Estimate of a Population Mean:s Unknown
Adequate Sample Size
If the population distribution is highly skewed or contains outliers, a sample size of 50 or more is recommended.
In most applications, a sample size of n = 30 is adequate when using the expression to develop an interval estimate of a population mean.
nstx 2/
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Interval Estimate of a Population Mean:s Unknown
Adequate Sample Size (continued)
If the population is believed to be at least approximately normal, a sample size of less than 15 can be used.
If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 will suffice.
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Summary of Interval Estimation Procedures
for a Population Mean
Can thepopulation standard
deviation s be assumed known ?
Use
Yes No
/ 2
sx t
nUse
/ 2x zn
s KnownCase
s UnknownCase
Use the samplestandard deviation
s to estimate s
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Let E = the desired margin of error.
E is the amount added to and subtracted from the point estimate to obtain an interval estimate.
Sample Size for an Interval Estimateof a Population Mean
If a desired margin of error is selected prior to sampling, the sample size necessary to satisfy the margin of error can be determined.
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Sample Size for an Interval Estimateof a Population Mean
E zn
/2
nz
E
( )/ 22 2
2
Margin of Error
Necessary Sample Size
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Sample Size for an Interval Estimateof a Population Mean
The Necessary Sample Size equation requires a value for the population standard deviation s .
If s is unknown, a preliminary or planning value for s can be used in the equation.
1. Use the estimate of the population standard deviation computed in a previous study.
2. Use a pilot study to select a preliminary study and use the sample standard deviation from the study.
3. Use judgment or a “best guess” for the value of s .
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Recall that Discount Sounds is evaluating apotential location for a new retail outlet, based
inpart, on the mean annual income of the
individuals inthe marketing area of the new location.
Sample Size for an Interval Estimateof a Population Mean
Example: Discount Sounds
Suppose that Discount Sounds’ management team
wants an estimate of the population mean such that
there is a .95 probability that the sampling error is
$500 or less.
How large a sample size is needed to meet the
required precision?
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At 95% confidence, z.025 = 1.96. Recall that = 4,500.
zn
/2 500
2 2
2
(1.96) (4,500)311.17 312
(500)n
Sample Size for an Interval Estimateof a Population Mean
A sample of size 312 is needed to reach a desired precision of + $500 at 95% confidence.
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The general form of an interval estimate of a population proportion is
Margin of Errorp
Interval Estimateof a Population Proportion
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Interval Estimateof a Population Proportion
The sampling distribution of plays a key role in computing the margin of error for this interval estimate.
p
The sampling distribution of can be approximated by a normal distribution whenever np > 5 and n(1 – p) > 5.
p
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/2/2 /2/2
Interval Estimateof a Population Proportion
Normal Approximation of Sampling Distribution of
p
Samplingdistribution of
Samplingdistribution of p
(1 )p
p p
n
ppp
/ 2 pz / 2 pz
1 - of all values1 - of all valuesp
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Interval Estimate
Interval Estimateof a Population Proportion
p zp pn
/
( )2
1
where: 1 - is the confidence coefficient z/2 is the z value providing an area of
/2 in the upper tail of the standard normal probability distribution
is the sample proportionp
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Political Science, Inc. (PSI) specializes in voter polls
and surveys designed to keep political office seekers
informed of their position in a race. Using telephone surveys, PSI interviewers
askregistered voters who they would vote for if the election were held that day.
Interval Estimateof a Population Proportion
Example: Political Science, Inc.
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In a current election campaign, PSI has just found
that 220 registered voters, out of 500 contacted, favor
a particular candidate. PSI wants to develop a 95%
confidence interval estimate for the proportion of the
population of registered voters that favor thecandidate.
Interval Estimateof a Population Proportion
Example: Political Science, Inc.
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or duplicated, or posted to a publicly accessible website, in whole or in part.
p zp pn
/
( )2
1
where: n = 500, = 220/500 = .44, z/2 = 1.96p
Interval Estimateof a Population Proportion
PSI is 95% confident that the proportion of all votersthat favor the candidate is between .3965 and .4835.PSI is 95% confident that the proportion of all votersthat favor the candidate is between .3965 and .4835.
.44(1 .44).44 1.96
500
= .44 + .0435
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Using Excel to Constructa Confidence Interval
Excel Formula WorksheetA B C
1 Favor Sample Size =COUNTA(A2:A501)2 Yes Response of Interest
=COUNTIF(A2:A501,C2)3 YesSample Proportion =C3/C14 No
5 YesConfid. Coefficient 0.956 No
Lev. of Significance =1-C67 Noz Value =NORM.S.INV(1-C7/2)8 No
9 NoStandard Error =SQRT(C4*(1-C4)/C1)10 YesMargin of Error =C8*C1011 No
12 YesPoint Estimate =C413 No
Lower Limit =C13-C1114 YesUpper Limit =C13+C1115
YesCount for Response
Yes
Note: Rows 16-501
are not shown.
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Using Excel to Constructa Confidence Interval
Excel Value WorksheetA B C
1 Favor Sample Size 5002 Yes Response of Interest
2203 YesSample Proportion4 No 0.4400
5 YesConfid. Coefficient 0.956 No
Lev. of Significance 0.057 Noz Value 1.96008 No
9 NoStandard Error 0.0222010 YesMargin of Error 0.0435111 No
12 YesPoint Estimate 0.440013 No
Lower Limit 0.396514 YesUpper Limit 0.483515
YesCount for Response
Yes
Note: Rows 16-501
are not shown.
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Solving for the necessary sample size, we get
Margin of Error
Sample Size for an Interval Estimateof a Population Proportion
/ 2
(1 )p pE z
n
2/ 2
2
( ) (1 )z p pn
E
However, will not be known until after we have selected the sample. We will use the planning valuep* for .
p
p
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Sample Size for an Interval Estimateof a Population Proportion
The planning value p* can be chosen by:1. Using the sample proportion from a
previous sample of the same or similar units, or
2. Selecting a preliminary sample and using the
sample proportion from this sample.3. Use judgment or a “best guess” for a p*
value.4. Otherwise, use .50 as the p* value.
2 * */ 2
2
( ) (1 )z p pn
E
Necessary Sample Size
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Suppose that PSI would like a .99 probability that
the sample proportion is within + .03 of thepopulation proportion. How large a sample size is needed to meet
therequired precision? (A previous sample of
similarunits yielded .44 for the sample proportion.)
Sample Size for an Interval Estimateof a Population Proportion
Example: Political Science, Inc.
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At 99% confidence, z.005 = 2.576. Recall that = .44.
p
A sample of size 1817 is needed to reach a desired precision of + .03 at 99% confidence.
2 2/ 2
2 2
( ) (1 ) (2.576) (.44)(.56) 1817
(.03)
z p pn
E
Sample Size for an Interval Estimateof a Population Proportion
/ 2
(1 ).03
p pz
n
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Note: We used .44 as the best estimate of p in thepreceding expression. If no information is availableabout p, then .5 is often assumed because it providesthe highest possible sample size. If we had usedp = .5, the recommended n would have been 1843.
Sample Size for an Interval Estimateof a Population Proportion
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End of Chapter 8