Chapter 8

# Chapter 8 - Ch 8 Inferences on Mean and Variance of a...

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Ch 8 Inferences on Mean and Variance of a Distribution

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2 8.1 Confidence Interval for Variance of a Normal Population Let X 1 , …, X n be a random sample from a normal distribution with mean and variance 2 . The random variable has a 2 distribution with n-1 degrees of freedom. 2 2 2 ) 1 ( s n
Deriving a 100(1-a)% Confidence Interval 1 ] ) 1 ( ) 1 ( [ 1 ] 1 ) 1 ( 1 [ 1 ] ) 1 ( [ 2 2 1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 2 2 2 2 1 s n s n P s n P s n P

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4 Confidence Interval Formula for Variance: 2 1 ], 2 [ 1 2 2 2 1 , 2 2 ) 1 ( ) 1 ( n n s n s n Confidence Interval for Standard Deviation: Find CI for variance and then take the positive square root of each limit.
5 Example: Find the 95% CI for a variance, using 15 degrees of freedom and s 2 = 10. 2 15 , 025 . 2 1 , 2 / n 2 15 , 975 . 2 1 , 2 / 1 n

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6 8.3 The Language of Hypothesis Testing A hypothesis is a statement or claim regarding a parameter of one or more populations. Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of one or more populations. The null hypothesis , denoted H 0 , is a statement to be tested. It will be a statement regarding the value of a population parameter. For this class, it will always be in the form of H 0 : parameter = some value. The alternative hypothesis, denoted H 1 or H A , is what we are trying to prove.
7 Parametric vs Nonparametric Hypothesis Testing Parametric These methods require that the data meet a specific set of assumptions: generally that the data is from a normal distribution or that the Central Limit Theorem applies. Nonparametric – Also known as “distribution - free” methods. These do not have a requirement of normality and are appropriate for smaller sample sizes as well.

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8 There are three ways to set up the alternative hypothesis. 1. “Not equal” ( two-tailed test ) H o: parameter = some value H 1: parameter does not equal some value 2. “Less than” ( one-tailed test ) H o: parameter = some value H 1: parameter < some value 3. “Greater than” ( one-tailed test ) H o: parameter = some value H 1: parameter > some value
9 Two possible decisions and conclusions: 1) Reject the Null Hypothesis: There is enough evidence to show that the alternative hypothesis is true. Interpretation: The sample evidence refutes the

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## This note was uploaded on 01/17/2011 for the course STAT 4714 at Virginia Tech.

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Chapter 8 - Ch 8 Inferences on Mean and Variance of a...

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