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DS212_Chap8_Stat_Inf_Single_Population

# DS212_Chap8_Stat_Inf_Single_Population - Chapter 8...

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1 Chapter 8: Statistical Inference: Estimation for Single Populations Sada Soorapanth Spring 2010

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2 Statistical Inference: s) (parameter , , Population p σ μ ) (statistic , , x Sample p s p p s , , estimate to , , x Calculate σ μ Select a random sample Process of Inferential Statistics
3 Statistical Estimation Point estimate -- the single value of a statistic calculated from a sample which is used to estimate a population parameter Interval Estimate -- a range of values within which the analyst can declare, with some confidence, the population parameter lies.

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4 Margin of Error and the Interval Estimate Each week JC department store selects a sample of 100 customers in order to learn about the amount spent per shopping trip (x). Based on historical data, the population standard deviation of amount spent was estimated at \$25. μ X 9 5 % of 9 . 4 96 . 1 = x σ X 4.9 4.9 ) 5 . 2 , ( ~ = x Normal x σ μ
5 100 × (1 - α )%Confidence Interval to estimate μ when σ is known Point estimate of μ Interval estimate of μ with 100(1- α )% confidence level n x x n i i = = 1 n z x n z x or n z x σ μ σ σ α α α 2 / 2 / 2 / + - ±

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6 What does a 95% Confidence Interval actually mean? μ X 95% X X X X X X
7 Distribution of Sample Means for 100(1- α )% Confidence Interval μ X 1 -α Z 0 α 2 Z - α 2 Z α 2 α 2

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8 Distribution of Sample Means for (1- α )% Confidence μ X Z 0 α 2 Z - α 2 Z α 2 α 2 .5 2 - α .5 2 - α
9 Example: Distribution of Sample Means for 99% Confidence Interval μ .495 .495 X 99% .005 .005 Z 2.576 -2.576 0

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10 Example: (from Black et al) A random sample of 81 items is taken, producing a sample mean of 47. The population standard deviation is 5.89. What is the point estimate of the population mean. Construct a 90% confidence interval to estimate the population mean.
11 Your Turn: (from Black et al) A random sample of size 70 is taken from a population that has a variance of 49. The sample mean is 90.4. What is the point estimate of μ ? Construct a 94% confidence interval for μ .

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12 Finite population correction factor 100 × (1 - α )%Confidence Interval to estimate μ for finite population (when σ is known) N = Population size 1 1 2 / 2 / - - + - - - N n N n z x N n N n z x σ μ σ α α
13 Example: (from Black et al) A random sample of size 39 is taken from a population of size 200 members. The sample mean is 66 and the population standard deviation is 11. Construct a 96% confidence interval to estimate the population mean. What is the point estimate of the population mean?

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14 100 × (1 - α )%Confidence Interval to estimate μ when σ is unknown Use the sample standard deviation s as a point estimate of σ .
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DS212_Chap8_Stat_Inf_Single_Population - Chapter 8...

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