chapter 7 estimating population values ©. chapter 7 - chapter outcomes after studying the material...
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Chapter 7Chapter 7
Estimating Population Estimating Population ValuesValues
©
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Chapter 7 - Chapter 7 - Chapter Chapter OutcomesOutcomes
After studying the material in this chapter, you should be able to:
•Distinguish between a point estimate and a confidence interval estimate.•Construct and interpret a confidence interval estimate for a single population mean using both the z and t distributions.
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Chapter 7 - Chapter 7 - Chapter Chapter OutcomesOutcomes
(continued)(continued)
After studying the material in this chapter, you should be able to:
•Determine the required sample size for an estimation application involving a single population mean.•Establish and interpret a confidence interval estimate for a single population proportion.
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Point EstimatesPoint Estimates
A point estimatepoint estimate is a single number determined from a sample that is used to estimate the corresponding population parameter.
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Sampling ErrorSampling Error
Sampling errorSampling error refers to the difference between a value (a statistic) computed from a sample and the corresponding value (a parameter) computed from a population.
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Confidence IntervalsConfidence Intervals
A confidence intervalconfidence interval refers to an interval developed from randomly sample values such that if all possible intervals of a given width were constructed, a percentage of these intervals, known as the confidence level, would include the true population parameter.
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Confidence IntervalsConfidence Intervals
Point EstimateLower Confidence
LimitUpper Confidence
Limit
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95% Confidence Intervals95% Confidence Intervals(Figure 7-3)(Figure 7-3)
0.95
z.025= -1.96 z.025= 1.96
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Confidence IntervalConfidence Interval- General Format -- General Format -
Point Estimate (Critical Value)(Standard Error)
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Confidence IntervalsConfidence Intervals
The confidence levelconfidence level refers to a percentage greater than 50 and less than 100 that corresponds to the percentage of all possible confidence intervals, based on a given size sample, that will contain the true population value.
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Confidence IntervalsConfidence Intervals
The confidence coefficient confidence coefficient refers to the confidence level divided by 100% -- i.e., the decimal equivalent of a confidence level.
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Confidence IntervalConfidence Interval- General Format: - General Format: known - known -
Point Estimate z (Standard Error)
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Confidence Interval Confidence Interval EstimatesEstimates
CONFIDENCE INTERVAL CONFIDENCE INTERVAL ESTIMATE FOR ESTIMATE FOR ( ( KNOWN) KNOWN)
where:z = Critical value from
standard normal table
= Population standard deviation
n = Sample size
nzx
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Example of a Confidence Example of a Confidence Interval Estimate for Interval Estimate for
A random sample of 100 cans, from a population with = 0.20, produced a sample mean equal to 12.09. A 95% confidence interval would be:
039.009.12100
20.096.109.12
n
zx
12.051 ounces
12.129 ounces
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Special Message about Special Message about Interpreting Confidence Interpreting Confidence
IntervalsIntervals
Once a confidence interval has been constructed, it will either contain the population mean or it will not. For a 95% confidence interval, if you were to produce all the possible confidence intervals using each possible sample mean from the population, 95% of these intervals would contain the population mean.
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Margin of ErrorMargin of Error
The margin of errormargin of error is the largest possible sampling error at the specified level of confidence.
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Margin of ErrorMargin of Error
MARGIN OF ERROR (ESTIMATE FOR MARGIN OF ERROR (ESTIMATE FOR WITH WITH KNOWN) KNOWN)
where:e = Margin of errorz = Critical value = Standard error of the
sampling distributionn
nze
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Example of Impact of Example of Impact of Sample Size on Sample Size on
Confidence IntervalsConfidence IntervalsIf instead of random sample of 100 cans, suppose a random sample of 400 cans, from a population with = 0.20, produced a sample mean equal to 12.09. A 95% confidence interval would be:
0196.009.12400
20.096.109.12
n
zx
12.051 ounces
12.129 ounces
12.0704 ounces
12.1096 ouncesn=400
n=100
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Student’s t-DistributionStudent’s t-Distribution
The t-distributiont-distribution is a family of distributions that is bell-shaped and symmetric like the standard normal distribution but with greater area in the tails. Each distribution in the t-family is defined by its degrees of freedom. As the degrees of freedom increase, the t-distribution approaches the standard normal distribution.
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Degrees of freedomDegrees of freedom
Degrees of freedomDegrees of freedom refers to the number of independent data values available to estimate the population’s standard deviation. If k parameters must be estimated before the population’s standard deviation can be calculated from a sample of size n, the degrees of freedom are equal to n - kn - k.
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t-Valuest-Values
t-VALUEt-VALUE
where:= Sample mean= Population mean
s = Sample standard deviation
n = Sample size
x
n
sx
t
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Confidence Interval Confidence Interval EstimatesEstimates
CONFIDENCE INTERVAL CONFIDENCE INTERVAL
(( UNKNOWN) UNKNOWN)
where:t = Critical value from t-
distribution with n-1 degrees of freedom
= Sample means = Sample standard deviationn = Sample size
n
stx
x
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Confidence Interval Confidence Interval EstimatesEstimates
CONFIDENCE INTERVAL-LARGE CONFIDENCE INTERVAL-LARGE SAMPLE WITH SAMPLE WITH UNKNOWN UNKNOWN
where:z =Value from the standard
normal distribution = Sample means = Sample standard deviationn = Sample size
n
szx
x
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Determining the Determining the Appropriate Sample SizeAppropriate Sample Size
SAMPLE SIZE REQUIREMENT - SAMPLE SIZE REQUIREMENT - ESTIMATING ESTIMATING WITH WITH KNOWN KNOWN
where:z = Critical value for the
specified confidence interval
e = Desired margin of error = Population standard
deviation
2
2
22
e
z
e
zn
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Pilot SamplesPilot Samples
A pilot samplepilot sample is a random sample taken from the population of interest of a size smaller than the anticipated sample size that is used to provide and estimate for the population standard deviation.
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Example of Determining Example of Determining Required Sample SizeRequired Sample Size
(Example 7-7)(Example 7-7)
The manager of the Georgia Timber Mill wishes to construct a 90% confidence interval with a margin of error of 0.50 inches in estimating the mean diameter of logs. A pilot sample of 100 logs yields a sample standard deviation of 4.8 inches.
Note, the manager needs only 150 more logs since the 100 in the pilot sample can be used.
25038.24950.0
)8.4(645.12
22
n
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Estimating A Population Estimating A Population ProportionProportion
SAMPLE PROPORTIONSAMPLE PROPORTION
where:x = Number of
occurrencesn = Sample size
n
xp
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Estimating a Population Estimating a Population ProportionProportion
STANDARD ERROR FOR STANDARD ERROR FOR pp
where: =Population
proportionn = Sample size
n
ppp
)1(
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Confidence Interval Confidence Interval Estimates for ProportionsEstimates for Proportions
CONFIDENCE INTERVAL FOR CONFIDENCE INTERVAL FOR
where:p = Sample proportionn = Sample sizez = Critical value from the
standard normal distribution
n
ppzp
)1(
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Example of Confidence Example of Confidence Interval for ProportionInterval for Proportion
(Example 7-8)(Example 7-8)
62 out of a sample of 100 individuals who were surveyed by Quick-Lube returned within one month to have their oil changed. To find a 90% confidence interval for the true proportion of customers who actually returned: 62.0
100
62
n
xp
100
)62.01)(62.0(645.162.0
0.50.544
0.70.700
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Determining the Required Determining the Required Sample SizeSample Size
MARGIN OF ERROR FOR ESTIMATINGMARGIN OF ERROR FOR ESTIMATING
where: = Population proportionz = Critical value from
standard normal distribution
n = Sample size
nze
)1(
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Determining the Required Determining the Required Sample SizeSample Size
SAMPLE SIZE FOR ESTIMATINGSAMPLE SIZE FOR ESTIMATING
where: = Value used to represent
the population proportion
e = Desired margin of errorz = Critical value from the
standard normal table
2
2 )1(
e
zn
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Key TermsKey Terms
• Confidence Coefficient
• Confidence Interval• Confidence Level• Degrees of Freedom• Margin of Error
• Pilot Sample• Point Estimate• Sampling Error• Student’s t-
distribution