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6.1 notes day 2 April 28, 2014 The Normal Distribution Positively skewed or rightskewed Symmetric mode, median, mean mean median mode Negatively skewed or leftskewed mean median mode The Normal Distribution "BellShaped" A normal distribution is a continuous, symmetric, bellshaped distribution of a variable 6.1 notes day 2 April 28, 2014 The Normal Distribution Not necessarily perfectly normally distributed, but approximately normally distributed. The shape and position of a normal distribution curve depends on 2 parameters: mean standard deviation Each normally distributed variable has its own normal distribution curve... same mean but different standard deviation 6.1 notes day 2 April 28, 2014 Each normally distributed variable has its own normal distribution curve... same standard deviation but different mean Each normally distributed variable has its own normal distribution curve... different standard deviation and different mean 6.1 notes day 2 April 28, 2014 Characteristics of a Normal Distribution 1. Bell shaped 2. mean,median,mode are equal 3. unimodal 4. The curve is symmetric about the mean 5. The curve is continuous 6. The curve never touches the x axis 7. The total area under the curve = 1 8. The area under the curve that lies within 1 standard deviation is about 68%, 2 standard deviations is 95% and 3 standard deviations is 99.7% 99.7% 95% 68% 6.1 notes day 2 April 28, 2014 34% 13.5% 2.35% 34% 13.5% 2.35% Standard Normal Distribution the standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1 6.1 notes day 2 April 28, 2014 Standard Normal Distribution 34% 34% 13.5% 13.5% 2.35% 2.35% Standard Normal Distribution All normally distributed variables can be transformed into the standard normally distributed variable by using the formula for the standard score. (z score) value mean standard deviation or z = 6.1 notes day 2 April 28, 2014 Standard Normal Distribution The z score is actually the number of standard deviations a value is away from the mean. You can use a table to find the area under the curve for any given z score. Finding Areas Under the Curve Given a z score/z scores, you can find the area under the curve. There are three types of problems you will see... 6.1 notes day 2 April 28, 2014 Finding Areas Under the Curve 1. Find the area under the curve to the left of a z score or Look up the z score in the table and use the given value. Finding Areas Under the Curve Find the area to the left of z = 1.99 1.99 z .00 .01 .02 .03 .04 .05 .05 .07 .08 .09 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761 .9767 6.1 notes day 2 April 28, 2014 Finding Areas Under the Curve 2. Find the area under the curve to the right of a z score or Look up the z score in the table and subtract the given value from 1. Finding Areas Under the Curve Find the area to the right of z = 1.16 1.16 6.1 notes day 2 April 28, 2014 Finding Areas Under the Curve 3. Find the area under the curve between z scores Look up the z scores in the table and subtract the corresponding areas. Finding Areas Under the Curve Find the area between z = 1.68 and z = 1.37 6.1 notes day 2 April 28, 2014 Probability Distribution Curves area under a normal distribution curve can be used as probabilities. 1.99 Notation: P(z<1.99) Probability Distribution Curves Find the probability of each P(0<z<2.32) P(z<1.65) P(z>1.91) 6.1 notes day 2 April 28, 2014 Finding Specific Z values We need to work backwards Find the z score such that the area under the standard normal distribution curve to the left of the z score is 0.5517 Finding Specific Z values We need to work backwards Find the z value such that the area under the standard normal distribution curve between 0 and the z value is 0.2123 0 z 6.1 notes day 2 April 28, 2014 Finding Specific Z values We need to work backwards Find the z value such that the area under the standard normal distribution curve to the right of the z value is 0.7123 z 0 Finding Specific Z values If the exact area cannot be found use the closest value!
