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Slides Prepared byJOHN S. LOUCKS

St. Edwards University

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Chapter 8Interval Estimation

Population Mean: Known Population Mean: Unknown Determining the Sample Size Population Proportion

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A point estimator cannot be expected to provide theexact value of the population parameter.

An interval estimate can be computed by adding andsubtracting a margin of error to the point estimate.

Point Estimate +/ Margin of Error

The purpose of an interval estimate is to provideinformation about how close the point estimate is tothe value of the parameter.

Margin of Error and the Interval Estimate

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The general form of an interval estimate of apopulation mean is

Margin of Errorx Margin of Errorx

Margin of Error and the Interval Estimate

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Interval Estimation of a Population Mean: 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 , or the sample standard deviation 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 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 /2z x /2

/2 /21 - of allvaluesxx

Samplingdistribution

of xx

xx

z x /2z x /2z x /2z x /2

Interval Estimation of a Population Mean: Known

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/2 /21 - of allvaluesxx

Samplingdistribution

of xx

xx

z x /2z x /2z x /2z x /2[------------------------- -------------------------]

[------------------------- -------------------------][------------------------- -------------------------]

xxxx

xx

intervaldoes notinclude interval

includes interval

includes

Interval Estimate of a Population Mean: Known

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Interval Estimate of

Interval Estimate of a Population Mean: Known

x zn

/2x z n

/2

where: is the sample mean1 - is the confidence coefficientz/2 is the z value providing an area of

/2 in the upper tail of the standard normal probability distribution

is the population standard deviationn is the sample size

xx

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Interval Estimate of a Population Mean: Known

Adequate Sample Size

In most applications, a sample size of n = 30 isadequate.

If the population distribution is highly skewed orcontains outliers, a sample size of 50 or more isrecommended.

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Interval Estimate of a Population Mean: Known

Adequate Sample Size (continued)

If the population is believed to be at leastapproximately normal, a sample size of less than 15can be used.

If the population is not normally distributed but isroughly symmetric, a sample size as small as 15 will suffice.

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SD

Interval Estimate of Population Mean: Known

Example: Discount SoundsDiscount Sounds has 260 retail outlets

throughout the United States. The firmis evaluating a potential location for anew outlet, based in part, on the meanannual income of the individuals inthe marketing area of the new location.

A sample of size n = 36 was taken;the sample mean income is $31,100. Thepopulation 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.

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95% of the sample means that can be observedare within + 1.96 of the population mean . xxThe margin of error is:

= =

/2

4,5001.96 1,47036

zn

= =

/24,5001.96 1,470

36z

n

Thus, at 95% confidence, the margin of erroris $1,470.

SD

Interval Estimate of Population Mean: Known

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Interval estimate of is:

Interval Estimate of Population Mean: Known

SD

We are 95% confident that the interval contains thepopulation mean.

$31,100 + $1,470or

$29,630 to $32,570

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Interval Estimation of a Population Mean: Unknown

If an estimate of the population standard deviation cannot be developed prior to sampling, we use the sample standard deviation s to estimate .

This is the unknown case. In this case, the interval estimate for is based on the

t distribution. (Well assume for now that the population is

normally distributed.)

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The t distribution is a family of similar probabilitydistributions.

t Distribution

A specific t distribution depends on a parameterknown as the degrees of freedom.

Degrees of freedom refer to the number of independent pieces of information that go into thecomputation of s.

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t Distribution

A t distribution with more degrees of freedom hasless dispersion.

As the number of degrees of freedom increases, thedifference between the t distribution and thestandard normal probability distribution becomessmaller and smaller.

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t Distribution

Standardnormal

distribution

t distribution(20 degreesof freedom)

t distribution(10 degreesof freedom)

0z, t

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For more than 100 degrees of freedom, the standardnormal z value provides a good approximation tothe t value.

t Distribution

The standard normal z values can be found in theinfinite degrees ( ) row of the t distribution table.

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t Distribution

Degrees Area in Upper Tailof Freedom .20 .10 .05 .025 .01 .005

. . . . . . .50 .849 1.299 1.676 2.009 2.403 2.67860 .848 1.296 1.671 2.000 2.390 2.66080 .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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Interval Estimate

x t sn

/2x tsn

/2

where: 1 - = the confidence coefficientt/2 = the t value providing an area of /2

in the upper tail of a t distributionwith n - 1 degrees of freedom

s = the sample standard deviation

Interval Estimation of a Population Mean: Unknown

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A reporter for a student newspaper is writing anarticle on the cost of off-campushousing. A sample of 16efficiency apartments within ahalf-mile of campus resulted ina sample mean of $650 per month and a samplestandard deviation of $55.

Interval Estimation of a Population Mean: Unknown

Example: Apartment Rents

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Let us provide a 95% confidence interval estimate of the mean rent permonth for the population of efficiency apartments within ahalf-mile of campus. We willassume this population to be normally distributed.

Interval Estimation of a Population Mean: Unknown

Example: Apartment Rents

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At 95% confidence, = .05, and /2 = .025.

Degrees Area in Upper Tailof Freedom .20 .100 .050 .025 .010 .005

15 .866 1.341 1.753 2.131 2.602 2.94716 .865 1.337 1.746 2.120 2.583 2.92117 .863 1.333 1.740 2.110 2.567 2.89818 .862 1.330 1.734 2.101 2.520 2.87819 .861 1.328 1.729 2.093 2.539 2.861. . . . . . .

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 Estimation of a Population Mean: Unknown

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x t sn

.025x tsn

.025

We are 95% confident that the mean rent per monthfor the population of efficiency apartments within ahalf-mile of campus is between $620.70 and $679.30.

Interval Estimate

Interval Estimation of a Population Mean: Unknown

55650 2.131 650 29.3016

= 55650 2.131 650 29.3016

=

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Summary of Interval Estimation Proceduresfor a Population Mean

Can thepopulation standard

deviation be assumed known ?

Use the samplestandard deviation

s to estimate s

Use

Yes No

/2sx tn

/2sx tn

Use

/2x z n

/2x z n

KnownCase

UnknownCase

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Let E = the desired margin of error.

E is the amount added to and subtracted from thepoint estimate to obtain an interval estimate.

Sample Size for an Interval Estimateof a Population Mean

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Sample Size for an Interval Estimateof a Population Mean

E zn

=

/2E z n=

/2

n zE

=( )/ 2

2 2

2nz

E=

( )/ 22 2

2

Margin of Error

Necessary Sample Size

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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 inthe marketing area of the new location.

Suppose that Discount Sounds management teamwants an estimate of the population mean such thatthere is a