Unformatted text preview: Chapter 8
Interval Estimation
• Population Mean: σ Known
• Population Mean: σ Unknown
• Determining the Sample Size
• Population Proportion Margin of Error and the Interval Estimate 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 The general form of an interval estimate of a population mean is x ± Margin of Error Interval Estimation of a Population Mean:
σ 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 σ , or
– the sample standard deviation s σ 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 σ known case. Interval Estimation of a Population Mean:
σ Known There is a 1 − α probability that the value of a
zα / 2 σ x
sample mean will provide a margin of error of or less.
Sampling distribution of x α/2 1 α of all
x values µ zα / 2 σ x α /2
x zα / 2 σ x Interval Estimate of a Population Mean:
σ Known
Sampling distribution of x α/2
interval
does not
include µ 1 α of all
x values zα / 2 σ x µ α/2 x [ x ] [[ x ] interval
includes µ zα / 2 σ x x ] Interval Estimate of a Population
Mean: σ Known
• Interval Estimate of µ
x ± zα / 2 σ
n where: is the sample mean
x 1 α is the confidence coefficient zα/2 is the z value providing an area of α/2 in the upper tail of the standard normal probability distribution σ is the population standard deviation n is the sample size Interval Estimate of a
Population Mean: σ 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. Interval Estimate of a
Population Mean: σ Known
• Adequate Sample Size (continued) If the population is not normally distributed but is roughly symmetric, a sample size as small as 15 will suffice. If the population is believed to be at least approximately normal, a sample size of less than 15 can be used. Interval Estimate of Population
Mean: σ Known
• Example: Discount Sounds Discount Sounds has 260 retail outlets throughout the United States. The firm
is evaluating a potential location for a
new outlet, based in part, on the mean
annual income of the individuals in
the marketing area of the new location. D S A sample of size n = 36 was taken;
the sample mean income is $31,100. The
population is not believed to be highly skewed. The population standard deviation is estimated to be $4,500, and the confidence coefficient to be used in the interval estimate is .95. Interval Estimate of Population
D
Mean: σ Known
S
95% of the sample means that can be observed
σx
are within + 1.96 of the population mean µ . The margin of error is: zα / 2 σ 4,500 = 1.96 = 1, 470
n 36 Thus, at 95% confidence, the margin of error is $1,470. Interval Estimate of Population
D
Mean: σ Known
S
Interval estimate of µ is:
$31,100 + $1,470
or
$29,630 to $32,570
We are 95% confident that the interval contains the
population mean. Interval Estimation of a
Population Mean: σ Unknown • If an estimate of the population standard deviation σ cannot be developed prior to sampling, we use the sample standard deviation s to estimate σ .
• This is the σ unknown case.
• In this case, the interval estimate for µ is based on the t distribution.
• (We’ll assume for now that the population is normally distributed.) t Distribution The t distribution is a family of similar probability distributions. 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. t Distribution A t distribution with more degrees of freedom has less dispersion. As the number of degrees of freedom increases, the difference between the t distribution and the standard normal probability distribution becomes smaller and smaller. t Distribution
t distribution
(20 degrees
of freedom) Standard
normal
distribution t distribution
(10 degrees of freedom)
z, t 0 t Distribution For more than 100 degrees of freedom, the standard normal z value provides a good approximation to the t value. The standard normal z values can be found in the infinite degrees ( ‹ ∞ ) row of the t distribution table. t Distribution
D egrees 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 normal
z values Interval Estimation of a Population Mean:
σ Unknown
• Interval Estimate
x ± tα / 2 s
n where: 1 α = the confidence coefficient tα/2 = the t value providing an area of α /2 in the upper tail of a t distribution with n 1 degrees of freedom s = the sample standard deviation Interval Estimation of a Population Mean:
σ Unknown • Example: Apartment Rents
A reporter for a student newspaper is writing an
article on the cost of offcampus
housing. A sample of 16
efficiency apartments within a
halfmile of campus resulted in
a sample mean of $650 per month and a sample
standard deviation of $55. Interval Estimation of a Population Mean:
σ 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
halfmile of campus. We will
assume this population to be normally distributed. Interval Estimation of a Population Mean:
σ Unknown
At 95% confidence, α = .05, and α /2 = .025.
t.025 is based on n − 1 = 16 − 1 = 15 degrees of freedom.
In the t distribution table we see that t.025 = 2.131.
D egrees Area in Upper Tail of Freedom .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 . .20 . . . . . . Interval Estimation of a Population Mean:
σ Unknown
s Interval Estimate x ± t.025 s
n Margin
of Error 55
650 ± 2.131
= 650 ± 29.30
16 We are 95% confident that the mean rent per month
for the population of efficiency apartments within a
halfmile of campus is between $620.70 and $679.30. Summary of Interval Estimation Procedures
for a Population Mean
Can the
population standard deviation σ be assumed known ? Yes Use the sample
standard deviation
s to estimate σ σ Known
Case
Use σ x ± zα / 2 n No σ Unknown
Case Use s x ± tα / 2 n Sample Size for an Interval Estimate
of a Population Mean 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 Estimate
of a Population Mean
s Margin of Error E = zα / 2
s σ
n Necessary Sample Size ( zα / 2 ) 2 σ 2
n=
E2 Sample Size for an Interval Estimate
of a Population Mean D Recall that Discount Sounds is evaluating a potential location for a new retail outlet, based in part, on the mean annual income of the individuals in
the marketing area of the new location.
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? S Sample Size for an Interval Estimate
of a Population Mean
zα / 2 D σ
= 500
n At 95% confidence, z.025 = 1.96. Recall that σ = 4,500. (1.96)2 (4, 500)2
n=
= 311.17 = 312
2
(500) A sample of size 312 is needed to reach a desired precision of + $500 at 95% confidence. S Interval Estimation
of a Population Proportion The general form of an interval estimate of a population proportion is p ± Margin of Error Interval Estimation
of a Population Proportion
p 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. Interval Estimation
of a Population Proportion
s Normal Approximation of Sampling Distribution of Sampling
distribution of p α /2 σp = 1 α of all
p values zα / 2 σ p p(1 − p )
n α/2 p p zα / 2 σ p p Interval Estimation
of a Population Proportion
• Interval Estimate
p (1 − p )
n p ± zα / 2 where: 1 α is the confidence coefficient zα/2 is the z value providing an area of α /2 in the upper tail of the standard normal probability distribution
p is the sample proportion Interval Estimation
of a Population Proportion
• Example: Political Science, Inc.
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 ask
registered voters who they would vote for if the
election were held that day. Interval Estimation
of a Population Proportion
• Example: Political Science, Inc.
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 the candidate. Interval Estimation
of a Population Proportion
p ± zα / 2 p (1 − p )
n where: n = 500, = 220/500 = .44, zα/2 = 1.96
p .44(1 − .44)
.44 ± 1.96
= .44 + .0435
500
PSI is 95% confident that the proportion of all voters
that favor the candidate is between .3965 and .4835. Sample Size for an Interval Estimate
of a Population Proportion
s Margin of Error p (1 − p )
n E = zα / 2 Solving for the necessary sample size, we get
( zα / 2 ) 2 p (1 − p )
n=
E2 p
However, will not be known until after we have selected the sample. We will use the planning value
p
p* for . Sample Size for an Interval Estimate
of a Population Proportion
s Necessary Sample Size ( zα / 2 )2 p* (1 − p* )
n=
E2
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. Sample Size for an Interval
Estimate
of a Population Proportion
Suppose that PSI would like a .99 probability
that the sample proportion is within + .03 of the population proportion.
How large a sample size is needed to meet the required precision? (A previous sample of similar units yielded .44 for the sample proportion.) Sample Size for an Interval Estimate
of a Population Proportion
zα / 2 p(1 − p )
= .03
n At 99% confidence, z.005 = 2.576. Recall that p .44.
= .44.
= n= ( zα / 2 )2 p(1 − p )
E2 (2.576)2 (.44)(.56)
=
≅ 1817
2
(.03) A sample of size 1817 is needed to reach a desired precision of + .03 at 99% confidence. Sample Size for an Interval Estimate
of a Population Proportion
Note: We used .44 as the best estimate of p in the
preceding expression. If no information is available
about p, then .5 is often assumed because it provides
the highest possible sample size. If we had used
p = .5, the recommended n would have been 1843. ...
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 Spring '09
 White
 Statistics, Normal Distribution, Standard Deviation, Estimation theory, t Distribution

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