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Class 4 - Linear Association

Class 4 - Linear Association - Confidence Interval Example...

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Confidence Interval Example n= 100 Ask 100 people how many watched NBA Playoffs last night Yes= 20 20 people said yes p^= 0.2 Want to find p-hat. What % watched NBA game SE(p-hat)= 0.04 Standard Error. Want to find the standard error of p-hat. CV(90%)= 1.645 Confidence interval or critical value. Add 5% tail to the 90% ME(90%)= 0.066 Margin of error. Critical Value x standard error Upper Limit= 0.266 Best estimate (p-hat) + Margin of error Lower Limit= 0.134 Best estimate (p-hat) - Margin of error
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The outcome or value of a statistic is a Random Variable and has a probability distribution. Since a statistic is computed from sample data its distribution is often referred to as the sampling distribution of a statistic . The standard deviation of the sampling distribution of a statistic is often referred to as the STANDARD ERROR of the statistic , SE(statistic).
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Variable Type Parameter Corresponding Sample Statistic Categorical Phenomenon Proportion p Sample Proportion (Location Parameter) Quantitative Phenomenon Mean μ Sample Mean (Location Parameter) Quantitative Phenomenon Variance Sample Variance (Spread Parameter) The 3 Sample Statistics are unbiased estimators of the corresponding σ 2 s 2 If a Statistic is on average = a Parameter then the Statistic is an unbiased estima Parameter p ˆ y 2 2 ) ( ) ( ) ˆ ( σ μ = = = s E y E p p E
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Std. Dev. Sample Theoretical Based Standard Standard Error Error Parameters Standard Errors are only used for Location Parameters. ator of the n p p ) 1 ( - n p p ) ˆ 1 ( ˆ - n σ n s
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Central Limit Theorem introduction If the distribution of values is normal then the distribution of the sample mean of n values from the distribution is also normal. What is the distribution of the sample mean if distribution of the individual values is not normal? The next two sheets allow examination of the distribution of the sample mean when Y follows a uniform and a triangular distribution. Changing the sample size n changes the shape of the distribution of the sample mean.
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n= 1 Mean = 100 Std. Dev. =100 SE = 100 Mean(y-bar) = 98.77 Skewness(y-bar)= 0.04 Std. Dev.(y-bar) = 97.67 Press the F9 key to obtain a new sample 1 bins Midpoin Frequency 2 1 ### 0 0 3 2 ### -64.09 24 26.31 4 3 ### -45.86 23 26.31 5 4 ### -27.63 31 26.31 10 1 =m 5 -0.28 -9.39 20 26.31 20 6 17.95 8.84 26 26.31 30 7 36.18 27.07 38 26.31 40 8 54.42 45.3 26.31 9 72.65 63.53 26.31 10 90.88 81.76 26.31 ### 26.31 ### 118.23 26.31 ### 136.46 26.31 ### 154.69 26.31 ### 172.92 26.31 ### 191.15 26.31 17 218.5 209.38 20 26.31 18 ### 227.62 11 26.31 19 ### 245.85 22 26.31 273.2 264.08 32 26.31 0 #N/A 473.64 -64.09 -45.86 -27.63 -9.39 8.84 27.07 45.3 63.53 81.76 100 118.23 136.46 154.69 172.92 191.15 209.38 22 0 5 10 15 20 25 30 35 40 / n = σ
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1 2 3 4 5 6 7 8 9 10 11 -49.11 ### ### ### ### ### ### ### ### -3.32 ### ### 44.07 ### ### ### ### ### ### ### ### ### -19.4 ### 205.01 ### ### ### ### ### ### ### ### ### ### ### 4.76 4.76 ### ### ### ### ### ### ### ### ### ### -36.73 ### ### ### ### ### ### ### ### -7.64 ### 9.01 167.91 ### ### ### ### ### ### ### ### ### ### ### -48.42 ### ### ### ### ### ### -39.1 ### ### ### ### -4.27 -4.27 ### ### -4.4 ### ### ### ### ### ### ### 252.25 ### ### ### ### 1.08 -9.37 ### ### ### ### ### 191.38 ### ### ### ### 79.5 ### ### ### ### ### ### 106.67 ### ### ### ### ### 71.4 ### ### ### ### ### -8.46 -8.46 ### 0.22 ### ### ### ### ### ### ### ### -63.37 ### ### ### ### ### ### ### ### ### ### ### 197.77 ### ### ### ### ### ### ### ### ### ### ### 108.48 ### ### ### 5.41 ### ### ### ### ### ### 2.22 153.82 ### ### ### ### ### ### ### ### ### ### ### -32.69 ### ### ### ### ### ### ### ### ### -6.8 ### 28.79 ### ### ### ### ### ### ### ### -8.95 ### ### 84.85 ### -26.2 8.61 2.42 6.41 ### 0.38 ### ### ### ### 215.38 ### ### 65.7 ### ### ### ### ### ### ### ### -12.2 -12.2 ### ### ### ### 42.3 ### ### ### ### ### 133.27 ### ### ### ### ### ### ### 9.61 3.48 ### ### 69.54 ### ### ### ### ### ### ### ### ### ### ### 259.1 ### -1.34 1.37 ### -38.9 ### ### ### ### ### ### 215.66 ### ### -0.91 ### ### ### 1.41 ### ### ### ### 248.86 ### -7.61 ### ### ### ### 21.1 -2.99 ### ### ### 198.98 ### ### ### ### ### ### ### ### ### ### ### 229.94 ### ### ### ### ### ### ### ### ### ### ### -13.34 ### 77.2 ### ### ### ### ### ### ### ### -1.29 1.03 1.03 ### -36.5 ### ### ### ### ### ### ### ### 33.21 ### ### ### ### ### ### ### ### ### ### ### 174.09 ### ### ### ### ### ### ### ### ### ### ### 61.51 ### ### ### ### ### ### ### ### ### 0.94 ### 38.93 ### -1.51 ### ### ### ### ### ### ### ### ### 85.1 85.1 ### ### ### ### ### ### ### ### ### -10.2 110.24 ### ### ### ### ### ### -6.86 ### ### ### ### 48.52 ### ### ### ### ### ### ### ### ### ### ### 260.4 ### ### ### ### ### ### ### ### ### ### 82 12.61 ### ### ### ### ### ### ### ### ### ### ### 56.15 ### ### ### ### ### ### ### ### ### ### -4.27 37.71 ### ### ### ### ### ### ### ### ### 96.8 ### 106.21 ### ### ### ### ### ### 3.13 ### ### ### ### 216.37 ### ### ### 146 ### ### ### 114 ### ### ### 27.62 245.85
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159.4 ### ### -4.22 ### ### ### ### ### ### ### ### -12.19 ### ### ### 78.1 ### ### ### 80.7 ### ### ### 41.57 ### ### ### ### ### ### ### ### ### ### 48.4 -46.72 ### ### ### ### ### ### ### ### ### ### ### -42.15 ### ### ### ### ### 38.4 ### ### ### ### ### 252.64 ### ### ### ### ### ### ### 1.25 ### ### ### 164.53 ### 5.9 -4.6 ### ### ### ### ### ### ### ### -31.9 -31.9 ### ### ### ### ### ### ### ### ### -4.6 -59.25 ### ### ### ### ### 43.6 ### ### ### ### ### 98.29 ### ### ### ### ### ### ### ### 1.58 ### ### 28.99 ### ### ### ### ### 103 ### ### ### ### ### 41.51 ### ### ### ### ### ### ### ### ### ### ### 24.24 ### ### ### ### ### ### ### ### ### ### -59.7 211.42 ### ### ### ### ### ### ### ### -56.7 -47.3 ### -34.85 ### 6.44 ### ### ### ### ### ### ### ### ### 122.37 ### ### ### ### ### ### ### ### ### ### ### 193.43 ### ### ### ### ### ### ### 139 ### ### ### 160.97 ### -9.34 ### ### -20.8 -3.49 ### ### ### ### ### -22.43 ### ### ### ### ### ### ### ### ### ### ### 71.81 ### 1.66 1.09 ### ### ### ### ### ### ### ### 19.83 ### ### ### ### ### ### ### ### ### ### ### 68.87 ### ### ### ### ### ### ### ### ### ### ### 72.54 ### ### ### ### ### ### ### ### ### ### ### 158.56 ### ### ### ### ### ### ### ### -7.89 ### ### 255.89 ### 114 ### ### ### ### ### ### 135 ### ### -70.08 ### ### ### ### ### 43.8 ### ### ### ### ### -61.55 ### ### ### ### ### ### ### ### ### ### -12.5 -51.13 ### ### ### -16.4 ### 57.1 ### ### ### ### -7.44 163.97 ### ### 7.28 ### -54.1 ### ### ### ### ### ### 175.66 ### ### ### ### ### -3.91 ### ### ### ### ### 135.08 ### ### ### ### ### 15.2 ### ### ### ### ### 39.63 ### ### ### ### ### ### 224 ### ### ### ### 78.37 ### ### ### ### ### ### ### ### ### ### ### 178.09 ### ### ### ### ### ### ### 1.28 1.49 ### ### 30.82 ### ### ### ### ### ### ### ### ### -8.51 ### 161.7 ### ### -4.89 ### ### ### ### ### ### ### ### 111.45 ### -62.9 ### ### ### ### ### ### ### 246 ### 194.31 ### ### ### ### ### ### ### ### ### -8.71 ### 270.07 ### ### ### ### ### ### ### ### ### ### ### 194.43 ### ### ### ### ### ### ### ### ### ### ### 186.46 ### ### ### 62.7 ### ### ### ### ### ### -25.7 123.33 ### ### ### ### ### -8.13 ### ### ### ### ### 260.13 ### ### ### ### ### ### ### ### ### 258 ### 218.17 ### ### ### ### ### ### ### ### ### ### ###
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235.74 ### ### ### ### ### -1.22 -34.8 ### ### ### ### 181.26 ### ### ### 159 ### 4.72 ### ### ### ### ### -33.54 ### ### ### ### ### ### ### ### ### ### -6.16 99.68 ### ### ### ### ### ### ### ### ### ### ### 198.14 ### ### ### ### ### ### ### ### ### ### ### 135.15 ### ### ### ### ### ### ### ### ### ### ### 70.04 ### ### ### ### ### ### ### ### ### ### ### 271.96 ### ### 1.71 ### ### 5.5 ### ### ### ### ### 254.4 ### ### ### ### ### 67.8 ### ### ### ### ### 58.7 58.7 ### ### ### ### ### ### ### ### ### ### 136.1 ### ### -3.28 ### ### ### ### ### ### ### -61.3 189.62 ### ### ### 1.69 ### ### ### 9.78 ### ### ### 112.98 ### -8.31 ### ### ### ### -4.68 ### ### ### ### 193.47 ### ### -9.25 ### ### ### ### ### ### ### ### -31.55 ### ### ### ### ### ### ### ### ### ### -72.9 -49.48 ### ### ### 69 ### ### ### ### ### ### ### 75.99 ### 6.42 ### ### ### ### ### ### ### ### ### 256.54 ### ### ### ### ### ### ### ### ### ### ### -40.78 ### ### ### ### ### ### -7.6 -1.27 ### ### ### -37.35 ### ### ### ### ### ### ### -3.7 -9.33 ### -52.4 232.15 ### ### ### ### ### ### ### ### ### ### ### 234.75 ### ### ### ### -3.17 ### ### ### ### ### ### -33.48 ### 35.5 ### ### ### ### ### ### ### ### ### 200.05 ### ### ### 3.8 -11 ### ### 228 ### ### ### 123.68 ### ### ### ### ### ### ### ### 0.72 ### ### -68.64 ### ### ### ### ### ### ### ### ### ### ### 172.15 ### ### ### ### -7.29 ### ### ### ### ### ### 16.54 ### ### ### -4.43 ### ### ### ### ### ### ### 60.71 ### ### ### ### ### ### ### ### ### ### ### 150.81 ### ### ### ### ### ### ### ### ### ### ### -1.4 -1.4 ### ### ### ### ### ### ### ### ### ### -47.2 -47.2 2.68 ### ### ### ### ### ### ### ### ### 240.08 ### ### ### ### ### ### ### ### -58.7 -2.6 ### 269.88 ### 88.7 ### -3.68 2.66 ### ### ### -11.7 ### ### 189.06 ### -2.37 6.62 ### ### ### ### ### 0.89 ### ### 254.03 ### ### 270 ### ### ### -8.94 ### ### ### ### 26.5 26.5 ### 179 ### ### ### ### ### ### ### 9.78 253.49 ### ### ### 4.81 ### ### -49.3 ### ### ### ### 13.39 ### ### ### ### ### ### ### ### ### ### ### 206.14 ### ### ### ### ### ### ### ### ### ### ### 72.21 ### -2.9 ### ### ### ### ### ### ### ### ### 159.42 ### ### ### ### ### ### ### ### ### ### ### 182.76 ### ### ### ### ### -3.59 ### ### ### 12.9 ### 173.95 ### -3.33 ### 241 ### ### ### ### ### ### ###
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