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Chapter 6: Probability

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Flow of inferential statistics and probability SamplePopulation Inferential Statistics Probability

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Jars of Marbles

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Probability Equation Probability of A = Number of outcomes classified as A Total number of possible outcomes

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Probability is Proportion Coin tosses -- p (heads) = ? Cards p (King of Hearts) = ? p (ace) = ? p (red ace) = ?

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Number list 6, 6, 7, 7, 7, 7, 8, 8, 8, 9

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Simple frequency table xf 91 83 74 62

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Random Sample 1.Each individual in the population has an equal chance of being selected. 2.If more than one individual is selected, there must be constant probability for each and every selection.

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Biased Sample

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1, 1, 2, 3, 3, 4, 4, 4, 5, 6 xf 61 51 43 32 21 12

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Frequency distribution 1 12345678 1 2 3 X frequency

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Frequency distribution 2 12345678 1 2 3 X frequency

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Normal Curve µ X

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Normal Curve with percentages 13.59% µ z -20+1+2 34.13% 13.59% 2.28%

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Comparison of Curves = 6 µ X 687480 6.5 (a) µ z 687480 (b) 0+2.00

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Curve broken down into + - 3 sd’s z -20+1+2-3+3

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Proportion in Tail vs. Body (A) z (B) Proportion in Body (C) Proportion in Tail 0.000.5000 0.010.50400.4960 0.020.50800.4920 0.030.51200.4880 0.210.58320.4168 0.220.58710.4129 0.230.59100.4090 0.240.59480.4052 0.250.59870.4013 0.260.60260.3974 0.270.60640.3936 0.280.61030.3897 0.290.61410.3859 Meanz z C C B

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Comparison of 3 curves (c) z 0 -.5 µ (a) 0 1.0 z µ (b) z 0 1.5 µ

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Comparison of 2 normal curves z (a) µ -0.4001.25 µ z (b) 0.3501.40

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Z-score formula in relation to probability XZ Probability z-score formula Unit normal table

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Normal curve with mean, 1 sd, and 2 z-scores = 100 X µ 500650 z 01.50

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Normal curve with mean, 2 sd’s, and 3 z-scores = 100 X µ 500700 z 02.00 600 1.00

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Normal curve with top 15% shaded = 100 X µ = 500 z 01.04 Top 15%

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Curve with mean of 100, shaded below 114 X µ = 100 z 01.40 = 10 114

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Curve with mean of 100, shaded below 92 X µ = 100 z 0-0.80 = 10 92

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Curve with mean of 60, shaded below 34% X µ = 60 = 5 P = 0.3400 or 34%

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Normal Curve with Quartiles 25% z Quartiles 0+0.67-0.67 Q1Q2Q3

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Probability of heads in two coin tosses 0.50 0.25 0 012 Probability Number of heads in two coin tosses

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Probability of heads in 4 & 6 coin tosses 01234 0.375 0.250 0.125 Probability (a) 0123456 Probability 0.2500 0.1875 0.1250 0. 0625 (b) Number of Heads in six coin tosses Number of Heads in four coin tosses

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The Relationship Between the Binomial Distribution and the Normal Distribution X 012345678910

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