Learn to recognize these symbols: sample mean population mean population standard deviation s sample standard deviation 1 5.1 Normal Distributions Transitioning from the Histogram to the "Density Curve" Histograms show total couts for certain class intervals..... whereas..... 2 Density Curves smooth out all the corners in order to make a continuous curve. 3 Histograms Have bars, so you can only find total counts for each class interval. But, Density Curves Are smooth, so that you are able to find proportions of some area under the curve. 4 Density Curves Always have a total area of one. 5 Density Curves allow us to find proportions under certain areas of the curve. These proportions can be expressed as percents, such as 4.95%. 6 Normal Density Curve The classic Bell Curve is perfectly Symmetrical, which means the MEAN and the STANDARD DEVIATION tell us virtually everything we need to know about the distribution The mean is always located at the 50th percentile (And, since it is symmetrical, so is the median incidently) 7 Percentiles Biff's IQ is at the 88th percentile This means that 88% of the IQ's are less than or equal to Biff's IQ. Or that 12% of the IQ's are higher than his. 8 9 The 68% 95% 99.7% Rule (Also called the Empirical Rule) For any given NORMAL distribution, there will always be a proportion 0.68 (68%) of the observations within one standard deviation of the mean. 10 The 68% 95% 99.7% Rule (Also called the Empirical Rule) For any given NORMAL distribution, there will always be a proportion 0.95 (95%) of the observations within two standard deviations of the mean. 11 The 68% 95% 99.7% Rule (Also called the Empirical Rule) For any given NORMAL distribution, there will always be a proportion of 0.99.7 (99.7%) of the observations within three standard deviations of the mean. 12 This is a very important and fundamental principle of Statistics and we will be applying it often throughout the rest of the year! 13 This will help you to answer many of the questions about how much percent is under some area of a normal density curve: • Copy the sketch of a normal distribution • Label the mean at the tick mark in the center • Label the tick marks that represent 1, 2, and 3 SD's above and below the mean .05 .06 .07 .08 .09 .1 .11 Now let's suppose that the BAC of male students at a particular college who drink 5 beers varies from student to student according to a normal distribution with mean 0.08 and standard deviation 0.01. • Label the estimated BAC levels on your diagram that would be represented at each of your tick marks 14 Classwork Practice 1. The middle 95% of students who drink 5 beers have BAC between what 2 numbers? 2. What percent of students who drink 5 beers have BAC above 0.08 (the legal limit for driving in most states)? 3. What percent of students who drink 5 beers have BAC above 0.10 (the legal limit for driving other states)? (e) 50 (d) 32% (c) 16% (b) 5% (a) 2.5% 4. What percent of students who drink 5 beers have BAC between 0.07 and 0.08? (e) 50% (d) 34% (c) 16% (b) 5% (a) 2.5% 15 16