1 1 slide interval estimation chapter 8 ba 201. 2 2 slide a point estimator cannot be expected to...
TRANSCRIPT
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1 1 Slide Slide
Interval Estimation
Chapter 8BA 201
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2 2 Slide Slide
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
A point estimator is a single statistic, e.g., the mean, that provides an estimate of a population parameter.
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3 3 Slide Slide
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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4 4 Slide Slide
Creating the Interval Estimate
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
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5 5 Slide Slide
INTERVAL ESTIMATE KNOWN
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6 6 Slide Slide
Interval Estimate of a Population Mean:s Known
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. In this case, the interval estimate for m is
based on the z distribution.
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7 7 Slide Slide
There is a 1 - probability that the value of asample mean will provide a margin of error of or less.
z x /2
1 - of all values1 - of all valuesx
/2/2 /2/2
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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8 8 Slide Slide
/2/2 /2/21 - of all values1 - of all valuesx
Sampling distribution of
Sampling distribution of x
x
z x /2z x /2
x[------------------------- -------------------------]
intervaldoes not
include m x[------------------------- -------------------------]
intervalincludes
mx[------------------------- -------------------------]
intervalincludes
m
Interval Estimate of a Population Mean:s Known
nx
Standard Error of the
Mean
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9 9 Slide Slide
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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10 10 Slide Slide
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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11 11 Slide Slide
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 0.90 is referred to as the confidence coefficient.
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12 12 Slide Slide
90% Confidence Interval
xx 645.1
xx 645.1 xx 645.1
5% greater than
5% less than
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13 13 Slide Slide
Comparison of Confidence Interval
90% CI
99% CI
Probability of not including is smaller, but CI is less precise.
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14 14 Slide Slide
Interval Estimate of a Population Mean: Known
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 0.95.
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15 15 Slide Slide
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
Discount Sounds
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16 16 Slide Slide
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
Discount Sounds
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17 17 Slide Slide
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
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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18 18 Slide Slide
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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19 19 Slide Slide
KNOWN PRACTICE
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20 20 Slide Slide
8-08
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21 21 Slide Slide
8-10
a) Develop a 90% confidence interval estimate of the population mean.b) Develop a 95% confidence interval estimate of the population mean.c) Develop a 99% confidence interval estimate of the population mean.
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22 22 Slide Slide
INTERVAL ESTIMATE UNKNOWN
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23 23 Slide Slide
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.
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24 24 Slide Slide
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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25 25 Slide Slide
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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26 26 Slide Slide
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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27 27 Slide Slide
t Distribution
Standardnormal
distribution
t distribution(20 degreesof freedom)
t distribution(10 degrees
of freedom)
0
z, t
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28 28 Slide Slide
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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29 29 Slide Slide
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 2.576
Standard normalz values
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30 30 Slide Slide
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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31 31 Slide Slide
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
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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32 32 Slide Slide
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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33 33 Slide Slide
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
7
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34 34 Slide Slide
UNKNOWN PRACTICE
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35 35 Slide Slide
8-12
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36 36 Slide Slide
8-16
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37 37 Slide Slide
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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38 38 Slide Slide
Summary of Interval Estimation Procedures
for a Population Mean
Can thepopulation standard
deviation s be assumed known ?
Yes
Use
/ 2x zn
s KnownCase
Use
No
/ 2
sx t
ns UnknownCase
Use the samplestandard deviation
s to estimate s
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39 39 Slide Slide