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PROBABILITY (6MTCOAE205). Chapter 6 Estimation. Confidence Intervals. Contents of this chapter: Confidence Intervals for the Population Mean, μ when Population Variance σ 2 is Known when Population Variance σ 2 is Unknown - PowerPoint PPT Presentation

### Transcript of Chapter 6 Estimation

Statistics for Business and Economics, 7/e

PROBABILITY (6MTCOAE205)Chapter 6

EstimationConfidence IntervalsContents of this chapter:Confidence Intervals for the Population Mean, when Population Variance 2 is Knownwhen Population Variance 2 is UnknownConfidence Intervals for the Population Proportion, (large samples)Confidence interval estimates for the variance of a normal population

Assist. Prof. Dr. mran Gker6-2DefinitionsAn estimator of a population parameter is a random variable that depends on sample information . . . whose value provides an approximation to this unknown parameter

A specific value of that random variable is called an estimate

Assist. Prof. Dr. mran Gker6-3Point and Interval EstimatesA point estimate is a single number, a confidence interval provides additional information about variabilityPoint EstimateLower Confidence LimitUpperConfidence LimitWidth of confidence intervalAssist. Prof. Dr. mran Gker6-4Point EstimatesWe can estimate a Population Parameter

with a SampleStatistic(a Point Estimate)MeanProportionP x

Assist. Prof. Dr. mran Gker6-5UnbiasednessA point estimator is said to be an unbiased estimator of the parameter if the expected value, or mean, of the sampling distribution of is ,

Examples: The sample mean is an unbiased estimator of The sample variance s2 is an unbiased estimator of 2The sample proportion is an unbiased estimator of P

Assist. Prof. Dr. mran Gker6-6

Unbiasedness is an unbiased estimator, is biased:

(continued)Assist. Prof. Dr. mran Gker6-7BiasLet be an estimator of

The bias in is defined as the difference between its mean and

The bias of an unbiased estimator is 0

Assist. Prof. Dr. mran Gker6-8Most Efficient EstimatorSuppose there are several unbiased estimators of The most efficient estimator or the minimum variance unbiased estimator of is the unbiased estimator with the smallest variance

Let and be two unbiased estimators of , based on the same number of sample observations. Then,

is said to be more efficient than if

The relative efficiency of with respect to is the ratio of their variances:

Assist. Prof. Dr. mran Gker6-9Confidence IntervalsHow much uncertainty is associated with a point estimate of a population parameter?

Such interval estimates are called confidence intervalsAssist. Prof. Dr. mran Gker6-10Confidence Interval EstimateAn interval gives a range of values:Takes into consideration variation in sample statistics from sample to sampleBased on observation from 1 sampleGives information about closeness to unknown population parametersStated in terms of level of confidenceCan never be 100% confidentAssist. Prof. Dr. mran Gker6-11Confidence Interval and Confidence LevelIf P(a < < b) = 1 - then the interval from a to b is called a 100(1 - )% confidence interval of .

The quantity (1 - ) is called the confidence level of the interval ( between 0 and 1)

In repeated samples of the population, the true value of the parameter would be contained in 100(1 - )% of intervals calculated this way. The confidence interval calculated in this manner is written as a < < b with 100(1 - )% confidenceAssist. Prof. Dr. mran Gker6-12Estimation Process(mean, , is unknown)PopulationRandom SampleMean X = 50SampleI am 95% confident that is between 40 & 60.Assist. Prof. Dr. mran Gker6-13Confidence Level, (1-)Suppose confidence level = 95% Also written (1 - ) = 0.95A relative frequency interpretation:From repeated samples, 95% of all the confidence intervals that can be constructed will contain the unknown true parameterA specific interval either will contain or will not contain the true parameterNo probability involved in a specific interval(continued)Assist. Prof. Dr. mran Gker6-14General FormulaThe general formula for all confidence intervals is:

The value of the reliability factor depends on the desired level of confidencePoint Estimate (Reliability Factor)(Standard Error)Assist. Prof. Dr. mran Gker6-15Confidence IntervalsPopulation Mean 2 UnknownConfidenceIntervalsPopulationProportion 2 KnownAssist. Prof. Dr. mran Gker6-16PopulationVarianceConfidence Interval for (2 Known) AssumptionsPopulation variance 2 is knownPopulation is normally distributedIf population is not normal, use large sampleConfidence interval estimate:

(where z/2 is the normal distribution value for a probability of /2 in each tail)

Assist. Prof. Dr. mran Gker6-17Margin of ErrorThe confidence interval,

Can also be written aswhere ME is called the margin of error

The interval width, w, is equal to twice the margin of error

Assist. Prof. Dr. mran Gker6-18Reducing the Margin of ErrorThe margin of error can be reduced if

the population standard deviation can be reduced ()

The sample size is increased (n)

The confidence level is decreased, (1 )

Assist. Prof. Dr. mran Gker6-19Finding the Reliability Factor, z/2Consider a 95% confidence interval:z = -1.96z = 1.96

Point EstimateLower Confidence LimitUpperConfidence LimitZ units:X units:Point Estimate0Find z.025 = 1.96 from the standard normal distribution tableAssist. Prof. Dr. mran Gker6-20Common Levels of ConfidenceCommonly used confidence levels are 90%, 95%, and 99%Confidence LevelConfidence Coefficient, Z/2 value1.281.6451.962.332.583.083.27.80.90.95.98.99.998.99980%90%95%98%99%99.8%99.9%

Assist. Prof. Dr. mran Gker6-21Intervals and Level of Confidence

Confidence Intervals Intervals extend from

to

100(1-)%of intervals constructed contain ; 100()% do not.Sampling Distribution of the Mean

xx1x2

Assist. Prof. Dr. mran Gker6-22ExampleA sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.

Determine a 95% confidence interval for the true mean resistance of the population.

Assist. Prof. Dr. mran Gker6-23ExampleA sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is .35 ohms. Solution:

(continued)

Assist. Prof. Dr. mran Gker6-24InterpretationWe are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean

Assist. Prof. Dr. mran Gker6-25Confidence IntervalsPopulation Mean 2 UnknownConfidenceIntervalsPopulationProportion 2 KnownAssist. Prof. Dr. mran Gker6-26PopulationVarianceStudents t DistributionConsider a random sample of n observationswith mean x and standard deviation s from a normally distributed population with mean

Then the variable

follows the Students t distribution with (n - 1) degrees of freedom

Assist. Prof. Dr. mran Gker6-27Confidence Interval for (2 Unknown) If the population standard deviation is unknown, we can substitute the sample standard deviation, s This introduces extra uncertainty, since s is variable from sample to sampleSo we use the t distribution instead of the normal distributionAssist. Prof. Dr. mran Gker6-28Confidence Interval for ( Unknown) AssumptionsPopulation standard deviation is unknownPopulation is normally distributedIf population is not normal, use large sampleUse Students t DistributionConfidence Interval Estimate:

where tn-1,/2 is the critical value of the t distribution with n-1 d.f. and an area of /2 in each tail:

(continued)

Assist. Prof. Dr. mran Gker6-29Margin of ErrorThe confidence interval,

Can also be written as

where ME is called the margin of error:

Assist. Prof. Dr. mran Gker6-30

Students t DistributionThe t is a family of distributionsThe t value depends on degrees of freedom (d.f.)Number of observations that are free to vary after sample mean has been calculatedd.f. = n - 1Assist. Prof. Dr. mran Gker6-31Students t Distributiont0t (df = 5) t (df = 13)t-distributions are bell-shaped and symmetric, but have fatter tails than the normalStandard Normal(t with df = )Note: t Z as n increasesAssist. Prof. Dr. mran Gker6-32Students t TableUpper Tail Areadf

.10.025.05112.706233.182t02.920The body of the table contains t values, not probabilitiesLet: n = 3 df = n - 1 = 2 = .10 /2 =.05/2 = .053.0781.8861.6386.3142.9202.3534.303Assist. Prof. Dr. mran Gker6-33t distribution valuesWith comparison to the Z valueConfidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ____

.80 1.372 1.325 1.310 1.282 .90 1.812 1.725 1.697 1.645 .95 2.228 2.086 2.042 1.960 .99 3.169 2.845 2.750 2.576Note: t Z as n increasesAssist. Prof. Dr. mran Gker6-34Example A random sample of n = 25 has x = 50 and s = 8. Form a 95% confidence interval for

d.f. = n 1 = 24, so

The confidence interval is

Assist. Prof. Dr. mran Gker6-35Confidence IntervalsPopulation Mean 2 UnknownConfidenceIntervalsPopulationProportion 2 KnownAssist. Prof. Dr. mran Gker6-36PopulationVarianceConfidence Intervals for the Population ProportionAn interval estimate for the population proportion ( P ) can be calculated by adding an allowance for uncertainty to the sample proportion ( )

Assist. Prof. Dr. mran Gker6-37Confidence Intervals for