Lec 03 - Tests for normal

# Lec 03 - Tests for normal - X P X P 8231 5 8231 5 8 1 =<...

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Is Data Normally Distributed? diam 11.5946 8.9121 9.6834 9.6545 8.0873 9.3021 5.8231 7.5982

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Histogram is Useless with only 8 observations! 2 diam Frequency 12 11 10 9 8 7 6 5 2.0 1.5 1.0 0.5 0.0 Mean 8.832 StDev 1.707 N 8 Histogram of diam Normal
Normal Probability Plot diam Percent 16 14 12 10 8 6 4 2 99 95 90 80 70 60 50 40 30 20 10 5 1 Mean 8.832 StDev 1.707 N 8 AD 0.231 P-Value 0.708 Probability Plot of diam Normal - 95% CI X axis: data points (in units of the data) Y axis: cumulative % of data points < X a normal probability scale 3

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How to Draw the Plot by Hand X axis is diameters Y axis is evenly spaced by standard deviations -2, -1, 0, 1, 2 Y axis is unevenly spaced by cumulative percent less than X .02, .16, .50, .84, .98 Step 1: Order points from low to high diam 11.5946 8.9121 9.6834 9.6545 8.0873 9.3021 5.8231 7.5982 4
Ordered from low to high (diam) 5.8231 7.5982 8.0873 8.9121 9.3021 9.6545 9.6834 11.5946 Calculate cumulative percent less than x: For each point i = (1), (2), ... (n) P i n i ( ) ( . ) = - 100 0 5 n=10 example: lowest out of 8 points is 5.8231 5

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6.25% or .5/8 - average correction point discrete

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Unformatted text preview: X P X P ) 8231 . 5 ( ) 8231 . 5 ( 8 1 = < = ≤ x value: 5.8231 y value: cumulative percent 6.25 use formula: P i n i ( ) ( . ) =-100 0 5 =5% try it for second point 6 Test of Hypothesis H : Data is normal with mean and sd as calculated from the sample Look at normal probability plot Do statistical tests: Ryan Joiner test: p>0.10 Anderson Darling test p=.708 Kolmogorov-Smirnov p>0.15 Conclusion: accept H • Shapiro Wilks/Ryan Joiner compares quantiles of observed data to normal distribution • Kolmogorov-Smirnov computes max distance between cumulative distribution function of data and of fitted normal 7 In a Perfect Data Set Suppose We Have Normal Data r.v. X distributed N(0,1) • cumulative probability fraction of data points ≤-2 .02 fraction of data points ≤-1 .16 fraction of data points ≤ .50 fraction of data points ≤ 1 .84 fraction of data points ≤ 2 .98 8...
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