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Transcript of © 2005 Thomson/South-Western Slide 1 - Governors · PDF file2009-01-05 · 6...

  • 1

    1Slide 2005 Thomson/South-Western

    Slides Prepared byJOHN S. LOUCKS

    St. Edwards University

    2Slide 2005 Thomson/South-Western

    Chapter 8Interval Estimation

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

    3Slide 2005 Thomson/South-Western

    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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    4Slide 2005 Thomson/South-Western

    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.

    6Slide 2005 Thomson/South-Western

    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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    7Slide 2005 Thomson/South-Western

    /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

    8Slide 2005 Thomson/South-Western

    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

    9Slide 2005 Thomson/South-Western

    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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    10Slide 2005 Thomson/South-Western

    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.

    11Slide 2005 Thomson/South-Western

    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.

    12Slide 2005 Thomson/South-Western

    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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    13Slide 2005 Thomson/South-Western

    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

    14Slide 2005 Thomson/South-Western

    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.)

    15Slide 2005 Thomson/South-Western

    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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    16Slide 2005 Thomson/South-Western

    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.

    17Slide 2005 Thomson/South-Western

    t Distribution

    Standardnormal

    distribution

    t distribution(20 degreesof freedom)

    t distribution(10 degreesof freedom)

    0z, t

    18Slide 2005 Thomson/South-Western

    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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    19Slide 2005 Thomson/South-Western

    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

    20Slide 2005 Thomson/South-Western

    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

    21Slide 2005 Thomson/South-Western

    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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    22Slide 2005 Thomson/South-Western

    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

    23Slide 2005 Thomson/South-Western

    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

    24Slide 2005 Thomson/South-Western

    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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    25Slide 2005 Thomson/South-Western

    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

    26Slide 2005 Thomson/South-Western

    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

    27Slide 2005 Thomson/South-Western

    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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    28Slide 2005 Thomson/South-Western

    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