Lecture_Statistics_Spring_2013b

Iii probability distribution and confidence interval

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Unformatted text preview: he effect of degrees of freedom on the Student's t-distribution and how it approaches a standard normal distribution as N → ∞. 0.4 Probability 0.3 z Distribution µ = 0, σ = 1 t Distributions ν = 30 ν=8 ν=3 0.2 0.1 0.0 -4 -2 0 t or z 2 4 See Figure 3 III. Probability Distributions & Confidence Intervals N observations (N-1 degrees of freedom) with the known experimental variance and mean a 100(1-α)% confidence interval is given as: ȹ s ȹ x ± t α / 2, υ = N −1 ȹ ȹ ȹ ȹ ȹ N Ⱥ ( ) Meaning of confidence interval: A 95% confidence interval means that 95% of all confidence intervals constructed according to the above equation will cover the true mean. III. Probability Distribution and Confidence Interval Weighted Average ȹ m ȹ ȹ m n i ȹ ȹ m ȹ x = ȹ ∑ n i x i ȹ N = ȹ ∑ x i ȹ = ȹ ∑ x i p(x i )ȹ ȹ ȹ ȹ N ȹ ȹ ȹ ȹ i =1 Ⱥ ȹ i =1 Ⱥ ȹ i =1 Ⱥ m: the number of unique observations and Ni: the number of times a given xi value is observed. *Weighted averages are important in chemistry Examples: atomic weights on a periodic table are weighted fro isotopes. IV. Propagation of Error How to determine the standard deviation of a multivariable function Q = f (X1, X2, ..., XN) comprised of N variables, X1, X2, X3, ….etc. 1) Take the total differential of this function: ȹ ∂Q ȹ ȹ ∂Q ȹ ȹ ∂Q ȹ ȹ ȹ ȹ ȹ dQ = ȹ dX1 + ȹ dX 2 + … + ȹ dX N...
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This note was uploaded on 01/26/2014 for the course CHEM 3625 taught by Professor Mrjohnson during the Spring '08 term at Virginia Tech.

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