Stat 101: Lecture 10 Normal Models • Our conceptualization of what the distribution of an entire population of values would look like. • Characterized by population parameters: µ and σ. 1 Sample Data 0.3 Density 0.2 0.1 0.0 85 90 95 100 Octane Rating 2 Normal Model 0.3 σ Density 0.2 0.1 0.0 85 90 µ 95 100 Octane Rating 3 Stat 101: Lecture 10 Normal Model • Octane Rating • Center: – Mean, µ = 91 • Spread: – Standard deviation, σ=1.5 4 68-95-99.7 Rule • 68% of the values fall within 1 standard deviation of the mean. • 95% of the values fall within 2 standard deviations of the mean. • 99.7% of the values fall within 3 standard deviations of the mean. 5 Octane Rating • 68% of the values fall between 89.5 and 92.5. • 95% of the values fall between 88.0 and 94.0. • 99.7% of the values fall between 86.5 and 95.5. 6 Stat 101: Lecture 10 Normal Percentiles • What percentage of octane ratings fall below 92? • Draw a picture. • How far away from the mean is 92 in terms of number of standard deviations? 7 Standard Normal Model • Table Z: in your text. • http://davidmlane.com/hyperstat/z_ table.html 8 From Percentiles to Scores • What octane rating corresponds to the 25th percentile? • Draw a picture. • The 25th percentile is how many standard deviations away from the mean? 9 Stat 101: Lecture 10 Are Your Data Normal? • The histogram should be mounded in the middle and symmetric. • The data plotted on a normal probability (quantile) plot should follow a diagonal line. 3 .99 2 .95 .90 1 .75 0 .50 .25 Normal Quantile Plot 10 -1 .10 .05 -2 .01 -3 6 4 Count 8 2 85 90 95 Octane Rating 3 .99 .95 .90 .75 .50 2 1 0 .25 .10 .05 .01 Normal Quantile Plot 11 -1 -2 -3 25 15 Count 20 10 5 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 Amplifier Gain (dB) 12