chapter 8 interval estimation population mean: known population mean: known population mean: ...
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Chapter 8Chapter 8Interval EstimationInterval Estimation
Population Mean: Population Mean: Known Known Population Mean: Population Mean: Unknown Unknown Determining the Sample SizeDetermining the Sample Size Population ProportionPopulation Proportion
A point estimator cannot be expected to provide theA point estimator cannot be expected to provide the exact value of the population parameter.exact value of the population parameter. A point estimator cannot be expected to provide theA point estimator cannot be expected to provide the exact value of the population parameter.exact value of the population parameter.
An An interval estimateinterval estimate can be computed by adding and can be computed by adding and subtracting a subtracting a margin of errormargin of error to the point estimate. to the point estimate.
An An interval estimateinterval estimate can be computed by adding and can be computed by adding and subtracting a subtracting a margin of errormargin of error to the point estimate. to the point estimate.
Point Estimate +/Point Estimate +/ Margin of Error Margin of Error
The purpose of an interval estimate is to provideThe purpose of an interval estimate is to provide information about how close the point estimate is toinformation about how close the point estimate is to the value of the parameter.the value of the parameter.
The purpose of an interval estimate is to provideThe purpose of an interval estimate is to provide information about how close the point estimate is toinformation about how close the point estimate is to the value of the parameter.the value of the parameter.
Margin of Error and the Interval EstimateMargin of Error and the Interval Estimate
The general form of an interval estimate of aThe general form of an interval estimate of a population mean ispopulation mean is
The general form of an interval estimate of aThe general form of an interval estimate of a population mean ispopulation mean is
Margin of Errorx Margin of Errorx
Margin of Error and the Interval EstimateMargin of Error and the Interval Estimate
Interval Estimation of a Population Mean:Interval Estimation of a Population Mean: Known Known
In order to develop an interval estimate of a In order to develop an interval estimate of a population mean, the margin of error must be population mean, the margin of error must be computed using either:computed using either:
• the population standard deviation the population standard deviation , or , or
• the sample standard deviation the sample standard deviation ss is rarely known exactly, but often a good is rarely known exactly, but often a good
estimate can be obtained based on historical estimate can be obtained based on historical data or other information.data or other information.
We refer to such cases as the We refer to such cases as the known known case. case.
There is a 1 There is a 1 probability that the value of a probability that the value of asample mean will provide a margin of error of sample mean will provide a margin of error of or less.or less.
z x /2z x /2
/2/2 /2/21 - of all values1 - of all valuesxx
Sampling distribution of
Sampling distribution of xx
xx
z x /2z x /2z x /2z x /2
Interval Estimation of a Population Mean:Interval Estimation of a Population Mean: Known Known
/2/2 /2/21 - of all values1 - of all valuesxx
Sampling distribution of
Sampling distribution of xx
xx
z x /2z x /2z x /2z x /2
[------------------------- -------------------------][------------------------- -------------------------]
[------------------------- -------------------------][------------------------- -------------------------]
[------------------------- -------------------------][------------------------- -------------------------]
xxxx
xx
intervalintervaldoes notdoes notinclude include
intervalintervalincludes includes
intervalintervalincludes includes
Interval Estimate of a Population Mean:Interval Estimate of a Population Mean: Known Known
Interval Estimate ofInterval Estimate of
Interval Estimate of a Population Mean:Interval Estimate of a Population Mean: Known Known
x zn
/2x zn
/2
where: is the sample meanwhere: is the sample mean 1 -1 - is the confidence coefficient is the confidence coefficient zz/2 /2 is the is the zz value providing an area of value providing an area of /2 in the upper tail of the standard /2 in the upper tail of the standard
normal probability distributionnormal probability distribution is the population standard deviationis the population standard deviation nn is the sample size is the sample size
xx
Interval Estimate of a Population Mean:Interval Estimate of a Population Mean: Known Known
Adequate Sample SizeAdequate Sample Size
In most applications, a sample size of In most applications, a sample size of nn = 30 is = 30 is adequate.adequate. In most applications, a sample size of In most applications, a sample size of nn = 30 is = 30 is adequate.adequate.
If the population distribution is highly skewed orIf the population distribution is highly skewed or contains outliers, a sample size of 50 or more iscontains outliers, a sample size of 50 or more is recommended.recommended.
If the population distribution is highly skewed orIf the population distribution is highly skewed or contains outliers, a sample size of 50 or more iscontains outliers, a sample size of 50 or more is recommended.recommended.
SSDD
Interval Estimate of Population Mean:Interval Estimate of Population Mean: Known Known
Example: Discount SoundsExample: Discount SoundsDiscount Sounds has 260 retail outlets Discount Sounds has 260 retail outlets
throughout the United States. The firmthroughout the United States. The firmis evaluating a potential location for ais evaluating a potential location for anew outlet, based in part, on the meannew outlet, based in part, on the meanannual income of the individuals inannual income of the individuals inthe marketing area of the new location.the marketing area of the new location.
A sample of size A sample of size nn = 36 was taken; = 36 was taken;the sample mean income is $31,100. Thethe sample mean income is $31,100. Thepopulation is not believed to be highly skewed. The population is not believed to be highly skewed. The population standard deviation is estimated to be $4,500,population standard deviation is estimated to be $4,500,and the confidence coefficient to be used in the interval and the confidence coefficient to be used in the interval estimate is .95.estimate is .95.
95% of the sample means that can be observed95% of the sample means that can be observed
are within are within ++ 1.96 of the population mean 1.96 of the population mean . . x xThe margin of error is: The margin of error is:
/ 2
4,5001.96 1,470
36z
n
/ 2
4,5001.96 1,470
36z
n
Thus, at 95% confidence, the margin of errorThus, at 95% confidence, the margin of error is $1,470. is $1,470.
SSDD
Interval Estimate of Population Mean:Interval Estimate of Population Mean: Known Known
Interval estimate of Interval estimate of is: is:
Interval Estimate of Population Mean:Interval Estimate of Population Mean: Known Known
SSDD
We are We are 95% confident95% confident that the interval contains the that the interval contains the
population mean.population mean.
$31,100 $31,100 ++ $1,470 $1,470oror
$29,630 to $32,570$29,630 to $32,570
Let Let EE = the desired margin of error. = the desired margin of error. Let Let EE = the desired margin of error. = the desired margin of error.
EE is the amount added to and subtracted from the is the amount added to and subtracted from the point estimate to obtain an interval estimate.point estimate to obtain an interval estimate. EE is the amount added to and subtracted from the is the amount added to and subtracted from the point estimate to obtain an interval estimate.point estimate to obtain an interval estimate.
Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean
Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean
E zn
/2E zn
/2
nz
E
( )/ 22 2
2n
z
E
( )/ 22 2
2
Margin of ErrorMargin of Error
Necessary Sample SizeNecessary Sample Size
Recall that Discount Sounds is evaluating a Recall that Discount Sounds is evaluating a potential location for a new retail outlet, based in potential location for a new retail outlet, based in part, on the mean annual income of the individuals part, on the mean annual income of the individuals ininthe marketing area of the new location.the marketing area of the new location.
Suppose that Discount Sounds’ management Suppose that Discount Sounds’ management teamteamwants an estimate of the population mean such thatwants an estimate of the population mean such thatthere is a .95 probability that the sampling error is there is a .95 probability that the sampling error is $500$500or less.or less.
How large a sample size is needed to meet How large a sample size is needed to meet thetherequired precision?required precision?
SSDD
Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean
At 95% confidence, At 95% confidence, zz.025.025 = 1.96. Recall that = 1.96. Recall that = 4,500.= 4,500.
zn
/2 500zn
/2 500
2 2
2
(1.96) (4,500)311.17 312
(500)n
2 2
2
(1.96) (4,500)311.17 312
(500)n
Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean
SSDD
A sample of size 312 is needed to reach a desiredA sample of size 312 is needed to reach a desired
precision of precision of ++ $500 at 95% confidence. $500 at 95% confidence.