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.

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.

Point EstimatesPoint Estimates

A point estimatepoint estimate is a single number determined from a sample that is used to estimate the corresponding population parameter.

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.

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.

Confidence IntervalsConfidence Intervals

Point EstimateLower Confidence

LimitUpper Confidence

Limit

95% Confidence Intervals95% Confidence Intervals(Figure 7-3)(Figure 7-3)

0.95

z.025= -1.96 z.025= 1.96

Confidence IntervalConfidence Interval- General Format -- General Format -

Point Estimate (Critical Value)(Standard Error)

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.

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.

Confidence IntervalConfidence Interval- General Format: - General Format: known - known -

Point Estimate z (Standard Error)

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

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

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.

Margin of ErrorMargin of Error

The margin of errormargin of error is the largest possible sampling error at the specified level of confidence.

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

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

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.

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.

t-Valuest-Values

t-VALUEt-VALUE

where:= Sample mean= Population mean

s = Sample standard deviation

n = Sample size

x

n

sx

t

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

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

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

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.

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

Estimating A Population Estimating A Population ProportionProportion

SAMPLE PROPORTIONSAMPLE PROPORTION

where:x = Number of

occurrencesn = Sample size

n

xp

Estimating a Population Estimating a Population ProportionProportion

STANDARD ERROR FOR STANDARD ERROR FOR pp

where: =Population

proportionn = Sample size

n

ppp

)1(

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(

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

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(

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

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