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Unformatted text preview: ESTIMATION AND INFERENCE THEORY • ESTIMATION OF PARAMETER • TESTING OF HYPOTHESIS POPULATION Population is the collection of observations about which conclusions are to be drawn SAMPLE Sample is a portion of population. PARAMETERS Parameters are statistical quantities of population. STATISTICS Statistics are statistical measures of sample INFERENCE THEORY Inference Theory deals with the method of drawing valid conclusions and prediction about the Population using information contained in the sample Topics to be covered ESTIMATION OF PARAMETER TESTING OF HYPOTHESIS ESTIMATION OF PARAMETERS ESTIMATION : Procedure of estimating a Population (PARAMETER) by using sample information is referred as estimation i) POINT ESTIMATION II) INTERVAL ESTIMATION POINT ESTIMATION POINT ESTIMATE : An estimate of a population parameter given by a single number is called point estimate POINT ESTIMATOR : A point estimator is a statistic for Estimating the population Parameter ө and will be denoted by ө * Example Problem of point estimation of the population mean µ : The statistic chosen will be called a point estimator for µ Logical estimator for µ is the Sample mean Hence µ* = ___ Χ _ _ _ Χ PROBLEM Market researcher use the number of sentences per advertisement as a measure of readability far magazine Advertisement. The following represents a random Sample of 54 advertisements. Find a point estimate of the population mean µ 9,20,18,16,9,16,16,9,11,13,22,16,5,18,6,6,5,12,25,17,23,7,10,9,10,10,5,11,18, 18,9,9,17,13,11,7,14,6,11,12,11,15,6, 12,14,11,4,9,18,12,12,17,11,20. Solution: The Sample mean of Data = 671/54 =12.4 So, Point Estimate for the mean length of all magazine advertisement is 12.4 sentences. UNBIASED ESTIMATOR Unbiased Estimator: If the mean of sampling distribution of a Statistic equals the corresponding Population Parameter,the Statistic is called an Unbiased Estimator of the Parameter i.e E( ө *) = ө Biased Estimator: If E( ө *) ≠ ө i.e Estimator is not Unbiased. Bias Of Estimator Bias of Estimator = E( *)  ө ө STANDARD ERROR OF THE MEAN Let denote the Sample mean based on a Sample of size n drawn from a distribution with standard deviation σ.The Standard deviation of is given by σ / and is called standard error of the mean ___ Χ ___ Χ n THEORM 1 The Sample mean is an unbiased estimate of the population mean Proof : Let x 1 ,x 2 ……..x n be a random sample of size n from a large population X 1 ,X 2 ,….,X N Of size N with mean µ . Sample mean is given by = ___ Χ ___ Χ ___ Χ ) ( / 1 ) 1 ( ) x E( Now 1 1 1 _ 1 i n i n i i n i i x E n n E n x x ∑ ∑ ∑ = = = = = Cont… Since x i is a sample observation from the population X i (i=1 to N) it can take any one of the values X 1 ,X 2 ,….X N each with equal propbability 1/N E ( x i ) = 1/N X 1 + 1/N X 2 +….. + 1/N X N = 1/N ( X 1 + X 2 +…..+ X N ) = µ _ Therefore E (x ) = 1/n ∑ (µ ) = 1/n ( n µ) = µ Hence the sample mean is an unbiased estimate of the population mean THEOREM 2:...
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 Spring '11
 ritadubey
 Normal Distribution, unbiased estimator

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