2.2 Interval Estimation - sigma unknown

# 2.2 Interval - Dr Harvey A Singer School of Management George Mason University OM210 Management 2 Confidence Interval Estimation of Means 2.2

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© 2008 by Harvey A. Singer 1 OM 210:  Statistical Analysis for  Management 2 Confidence Interval Estimation of Means 2.2 σ Unknown Dr. Harvey A. Singer School of Management George Mason University

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© 2008 by Harvey A. Singer 2 Remarks Up until now, have assumed that σ is known. And the previous confidence interval is exact. But often σ is not known, and has to be estimated from the standard deviation s of a sample. And the previous interval will be approximate. The quality of the approximation depends on the population distribution and the sample size.
© 2008 by Harvey A. Singer 3 Confidence Interval Estimates Confidence Intervals Mean Proportion Variance σ Unknown Finite Populations σ Known

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© 2008 by Harvey A. Singer 4 Confidence Interval for μ : σ Unknown Assumptions: Population standard deviation σ is unknown. But σ is estimated from the value of the sample standard deviation s . Population must be normally distributed. Still true that where E is the margin of error. But E is different because σ is unknown. Use s as the point estimate for σ . E x E - x + μ
© 2008 by Harvey A. Singer 5 Problems with σ Unknown Problems because σ is not known: Error from the accident of sampling (as before). Error using s as a first and best guess for σ . Error because of the finite size of the sample. Because of the finite sample size, there is the very strong likelihood that the sample is not representative of the population. As before, but magnified because s is used to estimate σ . A very big issue if the sample size is small ( n < 30).

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© 2008 by Harvey A. Singer 6 The Problems As a result, cannot use the normal distribution as a model for the sampling distribution of the sample mean. Need a new model for the sampling distribution of the sample mean. If the population being sampled is normal, then the new model is “Student’s” t - distribution.
© 2008 by Harvey A. Singer 7 Student’s t Distribution In appearance, similar to a normal curve. Centered on its mean. One central peak. Symmetrical about its mean. Asymptotes to the horizontal axis. But different. Shape depends on the size of the sample. According to the number of “degrees of freedom” df . For a sample of size n , df = n – 1.

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© 2008 by Harvey A. Singer 8 Student’s t Distributions z t 0 t for n = 6 ( df = 5) Standard Normal t for n = 14 ( df = 13) Bell-Shaped Symmetric “Fatter” Tails
© 2008 by Harvey A. Singer 9 Student’s t Distributions Comparisons:

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© 2008 by Harvey A. Singer 10 Student’s t values Two parameters for a t value, t α /2, n – 1 : The confidence level 1 α . α
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## This note was uploaded on 01/26/2011 for the course OM 210 taught by Professor Singer during the Fall '08 term at George Mason.

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2.2 Interval - Dr Harvey A Singer School of Management George Mason University OM210 Management 2 Confidence Interval Estimation of Means 2.2

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