C1000__19[1]

# C1000__19[1] - The normal approximation to the Binomial...

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1 The normal approximation to the Binomial variable N(μ=np,σ =np(1-p)) B(n,p)

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2 0 1 2 3 4 5 6 In a large bowl of M&M’s, the proportion of blues is 1/6 (or .17). X~B(6, 1/6) Draw the probability histogram of X and compute its mean and SD x p(x) 0 1(1/6) 0 (5/6) 6 =.33 1 6(1/6) 1 (5/6) 5 =.40 2 15(1/6) 2 (5/6) 4 =.20 3 20(1/6) 3 (5/6) 3 =.05 4 15(1/6) 4 (5/6) 2 =.01 5 6(1/6) 5 (5/6) 1 =.00 6 1(1/6) 6 (5/6) 0 =.00 .5 .4 .3 .2 .1 Shape: Skewed right Mean: 6(1/6)=1 SD: √6(1/6)(5/6)=.9
3 X~B(30, 1/6) Describe the center, spread, and shape of the distribution of X x P( X = x ) 0.00 0.0037 1.00 0.0230 2.00 0.0682 . . . 30.00 0.000 Minitab: ..\binomial in class.MPJ

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4 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0.2 0.1 0.0 X p(x) Shape: Smoother, more bell shaped Mean: 30(1/6)=5 SD: √30(1/6)(5/6)=2 X~B(30, 1/6)
5 X~B(90, 1/6) Describe the center, spread, and shape of the distribution of X x P( X = x ) 0.00 0.0000 1.00 0.0000 2.00 0.0000 3.00 0.0001 4.00 0.0002 . . 90.00 0.000

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6 90 89 88 87 86 85 84 83 82 81 80 79 78 77 76 75 74 73 72 71 70 69 68 67 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0.10 0.05 0.00 X p(x) Shape: Even smoother, more bell shaped – very close to a normal curve Mean: 90(1/6)=15 SD: √90(1/6)(5/6)=3.5 X~B(90, 1/6)
7 The binomial variable X~B(90, 1/6) behaves approximately like a normal variable with mean 15 and SD 3.5

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C1000__19[1] - The normal approximation to the Binomial...

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