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Relationship Between Sample Data and Population Values You will encounter many situations in business where a sample will be taken from a population, and you will be required to analyze the sample data. Regardless of how careful you are in using proper sampling methods, the sample likely will not be a perfect reflection of the population. Sampling Distribution A Sampling Distribution is the probability distribution for a statistic. Its description includes: • all possible values that can occur for the statistic; and • the probability of each value or each interval of values for a given sample. Example Individual A B C D E Annual Income $50,000 45,000 15,000 38,000 22,000 170,000 ( Income - µX) 6 256*10 121*106 6 361*10 6 16*10 144*106 6 898*10 2 Population Parameters • Population Mean (µX): µX = 170,000 / 5 = $34,000 • Population Standard Deviation (X): X = [SQRT(898*106) / 5] = $13,401.49 Draw a Random Sample of Three • How many random samples of three can you draw from this population? 5C3 = 10 samples of three can be drawn form this population. Each sample has a 1 / 5C3 , or 1 / 10 chance of being selected. • List the sample space and find sample means. Ten Possible Samples Sample A, B, C A, B, D A, B, E A, C, D …. C, D, E Income Levels Sample Mean ($1,000) 50, 45, 15 $36,666.67 50, 45, 38 44,333.33 50, 45, 22 39,000.00 50, 15, 38 34,333.33 …. …. 15, 38, 22 25,000.00 The Sampling Distribution of Sample Means ( X ) • The mean of the samples means: µX = ( X1 + X2 + …. + Xn ) / NCn µX = 340,000 / 10 = $34,000 • The Standard Deviation the samples means, better known as the Standard Error of the Mean: X = SQRT[( Xi - µX )2 / NCn] Standard Error of the Mean • The standard error of the mean indicates the spread in the distribution of all possible sample means. • X is also equal to the population standard deviation divided by the SQRT of the sample size X = X / SQRT(n) A Finite Population Correction Factor (fpc) • For n > 0.05N, the finite population correction factor adjusts the standard error to most accurately describe the amount of variation. • The fpc is SQRT[( N - n ) / ( N - 1 )]