QM-1/2011/EstimationPage 1

Quantitative MethodsEstimation

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Estimation Process

PopulationPopulation

Mean, , is unknown

Random SampleRandom Sample

Mean X= 50 Are we

confident that is

50?

OR we are more confident in saying that is between 48

& 52?

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Population Parameters are Estimated

Population Parameter Sample Statistic

2

xi

n

(xi-x)2

n-1

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Estimation Methods

Estimation

Point Estimation Interval Estimation

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Estimation Methods

Estimation

Point Estimation Interval Estimation

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Point Estimation Point Estimate

Draw a sample from a population and compute AVERAGE to ESTIMATE the MEAN Value of the population.

The formula used for computing Average is called ‘Statistic’ or ‘Estimator’.

The computed value based on the sample is called ‘Estimate’ for the population parameter (here ‘Mean’).

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Problem with Point Estimate Following data is a sample from a

population. 13,13,16,19,17,15,14,13,15,15,18. Point estimate for Mean is simply the

average of these data, i.e. 15.27. If we had collected lesser or larger

number of samples the average would have been different.

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Problem with Point Estimate

Population Parameter

1st groupof sample

2nd groupof sample

Estimates from different samples will be different,

but, within a limit, around the Population

parameter. This is applicable for all estimates (i.e.

Average, s.d. etc). All estimates from different

samples are ‘Point estimates’.

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Estimation Methods

Estimation

Point Estimation Interval Estimation

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Interval Estimate This estimate gives a lower and upper

limit. The population parameter will be in this interval with a certain degree of confidence.A confidence level is associated with an

interval estimate.Higher the confidence level desired, the

spread of the interval will be larger.

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Error in Estimate

Sample statistic Sample statistic

(point estimate)(point estimate)

Population Parameter Population Parameter (unknown)(unknown)

x

X ~ N(, /√n)

~ N(0, 1)X – /√n

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Standard Normal & Tail area

0.95 0.05

Z 0.05

-4 -3 -2 -1 0 1 2 3 4

P(Z < Z0.05) = 0.95

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Standard Normal & Tail area

(1- )

Z

-4 -3 -2 -1 0 1 2 3 4

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Standard Normal & 2-Tail area

-4 -3 -2 -1 0 1 2 3 4

(1- )

Z

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Estimation of Interval P(|Z |< Z0.025) = 0.95

P(|X – | < Z0.025) = 0.95

P(|X – | < Z0.025 /√n) = 0.95

P(-Z0.025 /√n < - X < Z0.025 /√n) = 0.95

P(X - Z0.025 /√n < < X + Z0.025 /√n) = 0.95

/√n

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Confidence Interval for Mean P(X – Z/2 /√n < < X + Z/2 /√n) = 1-

(1-) /2 Z/2

0.80 0.20 0.10 1.28

0.90 0.10 0.05 1.65

0.95 0.05 0.025 1.96

0.98 0.02 0.01 2.33

0.99 0.01 0.005 2.58

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Wider interval for higher confidence

99%99%

xx - 2.58- 2.58xxxx + 2.58+ 2.58xx

95%95%

x -1.96x -1.96xx x +1.96x +1.96xx

90%90%

x -1.65x -1.65xx x +1.65x +1.65xx

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Factors Affecting Interval Width Data Dispersion

Measured by Higher results wider interval.

Sample SizeX = n

Higher sample size results narrower interval.

Level of Confidence (1 - )Affects ZHigher level of confidence, wider interval.

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Confidence Interval Estimates

Confidence Interval

Mean Proportion

Known Unknown

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Confidence Interval Estimates

Confidence Interval

Mean Proportion

Known Unknown

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Confidence Interval of Mean - known

Population Standard Deviation is Known Population is Normally Distributed. n > 30, if Population is Not Normal. Confidence Interval:

nZX

nZX

2/2/

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Exercise The mean of a random sample of n = 36

isX = 50. Set up a 95% confidence interval estimate for if = 12.

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Exercise You’re a Q/C inspector of Coke. The for

2-liter bottles is 0.05 liters. A random sample of 100 bottles showedX = 1.99 liters. What is the 90% confidence interval estimate of the true mean amount in 2-liter bottles?

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Confidence Interval Estimates

Confidence Interval

Mean Proportion

Known Unknown

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Confidence Interval of Mean - Unknown

Normal population. n > 30 The sample sd s is a good estimator of Confidence interval is as below.

nZX

nZX

ss 2/2/