Chap 10 - Sampling & Estimation

Chap 10 - Sampling & Estimation
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  BUSINESS STATISTICS (DCA 102)CHAPTER 10 : SAMPLING AND ESTIMATIONPREPARED BY : MONA SEOWPage 1 of 5 CHAPTER TEN : SAMPLING AND ESTIMATIONIntroduction ã   To learn the population characteristics we have to use samples. A sample is aproportion or sub-set of the population, and is used to predict the characteristics of the population. Based on the principle of statistical regularity and the principle of “Inertia of large numbers”, if the collected sample is not biased and is large enough,to a certain extent the sample results will reveal the characteristics of thepopulation. ã   Population parameters can be predicted from samples. We can collect a samplefrom the population and compute appropriate sample statistics. The sample statisticsis used to predict the value of the population parameters. This is known ase stimation.    Estimation  can be defined as “ techniques used to establish a value foran unknown population parameter using the respective sample statistics”. PopulationSampleCensusRandom SamplingAttribute SamplingPopulation Parameter  is the value obtained from a set of data which represents all theobservations in the designated population. Sample Statistic  is the value that describes the sample.The table below shows the designated statistical value, the symbol used to describe asample and a population.Symbols UsedStatistical value Sample statisticPopulation parameter Mean  x  µ Standard Deviation s  σ Variance s 2 σ 2 Proportion  p  π Standard Error (SE) Standard error measures the sampling error associated with a sampling distribution. Thelarger the size of the sample taken, the smaller the standard error.  BUSINESS STATISTICS (DCA 102)CHAPTER 10 : SAMPLING AND ESTIMATIONPREPARED BY : MONA SEOWPage 2 of 5 SE for sampling distribution of the mean nSE   σ  =  where σ : population standard deviation and n : sample size usedIf ‘ σ ’ is not known, then the sample standard deviation , s , can be used as follows: nsSE   = If the population is finite, with the size  N  , then the standard error would be:1. −−=  N n N nsSE  SE for sampling distribution of the proportion n p pSE  )1(  −=  where  p : sample proportionIf the population is finite, with the size  N  , then the standard error would be:1.)1( −−−=  N n N n p pSE  Central Limit Theorem The Central Limit Theorem states that “as the sample size increases, the samplingdistribution of the sample mean approaches the normal distribution in shape, regardlessof the shape of the population”.For samples taken from a non-normal population with mean µ  and variance σ 2 , by theCentral Limit Theorem, _  X   is approximately normal. Point Estimate A point estimate for a population parameter is the value of the sample statistic that isdirectly used to estimate the population parameter.  BUSINESS STATISTICS (DCA 102)CHAPTER 10 : SAMPLING AND ESTIMATIONPREPARED BY : MONA SEOWPage 3 of 5 Example 1A survey was conducted to determine the average amount of money spent weekly bystudents in a particular college. Data collected from the sample of 40 students (20 maleand 20 female students) are given below:25, 15, 30, 40, 25, 20, 25, 40, 20, 25, 50, 25, 20, 15, 50,30, 20, 20, 25, 30, 35,40, 100, 50, 25, 20, 25, 25, 40, 100, 30, 20, 20, 25, 30, 20, 20, 25, 50, 30The mean amount of money spent = RM32. This value is directly used to make aconclusion about the mean amount of money spent by all the students in the college.Hence, using the point estimate, we say that the mean amount of money spent by studentsin the college (population) is RM32 . Interval Estimate An interval estimate of a population parameter consists of an interval of numbersobtained from a point estimate of the population parameter together with a percentagethat specifies how confident we are that the population parameter lies in the interval. Population parameter = Sample statistic ±   Sampling Error Population parameter = Sample statistic ±   confidence limit x  standard error  Confidence interval95% (0.95)99% (0.99) α  = 1 – confidence interval 0.050.01Two sided test ( α  /2) 0.0250.005Estimating the population mean for a large sampleA sample is said to be a large sample if its size is equal to or greater than 30. A sample of this size, based on the central limit theorem, can be said to be normally distributed.  N  xdarderror sit confidence x σ  µ  µ  α  2 tanlim Ζ ±=×±=  BUSINESS STATISTICS (DCA 102)CHAPTER 10 : SAMPLING AND ESTIMATIONPREPARED BY : MONA SEOWPage 4 of 5 Estimating the population mean for a small sampleIf the sample size, n , is small, i.e. n<30 , then the t-distribution is used to estimate thepopulation parameters.  N t  xdarderror sit confidence x n σ  µ  µ  α  1,2 tanlim − ±=×±= degrees of freedom , v = n – 1 Estimating the population proportionThe population proportion can be calculated using the following mathematical equation: n p p pror satndarder it confidence p )1(lim 2 −Ζ ±=×±= α  π π  Determining the sample size By increasing the sample size, we can reduce the sampling error. Sampling error, r = confidence interval ns × Population MeanPopulation Proportion 2      === r snr snnsr  2222 )1()96.1( )1()96.1( )1(96.1 r  p pnn p pr n p pr  −=−=−=
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