Stats LEC 3 - Common Language and Notations Parameters and...

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Parameters and Statistics Variables can be summarized using statistics. • A statistic is a numerical measure that describes a characteristic of the sample •A parameter is a numerical measure that describes a characteristic of the population. • We use statistics to estimate parameters Common Language and Notations:
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A population is an entire group of which we want to characterize. Population parameters: mean, variance, standard deviation, proportion. A sample is a collection of observations on which we measure one or more characteristics. Sample statistics: mean, variance, standard deviation, proportion. Population Sample
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Population Sample Notations: Estimation and Inferences n i i 1 n 2 2 i i 1 2 i Population Parameters : 1 Mean x N 1 Var (x ) N (x ) St. Dev. N proportion p μ ! μ μ ! = = = = " " = # # # Sample Statistics : Mean x = 1 n x i i = 1 n ! Var s 2 = 1 n " 1 (x i " x) 2 i = 1 n ! St. Dev. s = (x i " x) 2 ! n " 1 proportion ˆ p
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The Normal Distribution
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The Normal Distribution Bell Shaped Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: + to Mean = Median = Mode x f(x) μ σ
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By varying the parameters μ and σ , we obtain different normal distributions Many Normal Distributions
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The Normal Distribution Shape x f(x) μ σ Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. Given the mean μ and variance σ we define the normal distribution using the notation X~N( μ , ! )
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The formula for the normal distribution is Where e = the mathematical constant approximated by 2.71828 π = the mathematical constant approximated by 3.14159 μ = the population mean σ = the population standard deviation x = any value of the continuous variable, −∞ < x < f(x) = 1 2 ! !
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