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Probability and Statistics Normal Distributions Chapter 6 Section 3 Area Under Any Normal Curve Essential Question: What is the process for determining the area under a normal curve? Student Objectives: The students will calculate the probability of a “standardized” event. The students will find a z-score given the normal probability (inverse normal). The student will use the inverse normal to solve a guarantee problem. Terms: Inverse normal Pearson’s Index Standard Normal Curve z-score Key Concepts: Converting any normally distribution into a standard normal distribution. This allows the use of one table of values to determine the all the probabilities. Check for Normality 1. Draw a histogram. For a normal distribution, the histogram should be roughly bell-shaped. 2. Outliers. for a normal distribution, there should be no more than one outlier on a box-n-whisker plot. 3. Skewness. Normal distributions are symmetric. Use Pearson’s index for skewness. A score smaller than negative 1 or larger than 1 indicates the data is skewed. 4. Normal quartile plot. This graph plots points as (z-score, x-value). The data is considered NOT to be skewed if the points lie in a straight line. Equations: z-score: z = x !µ " Pearson's Index of Skewness !1 " 3( x ! median ) "1 s Graphing Calculator Skills: 2nd VARS DISTR Normalcdf(lower bound, upper bound, µ , ! ) 2nd VARS DRAW ShadeNorm(lower bound, upper bound, µ , ! ) 2nd VARS DISTR invNorm(area, µ , ! ) 2nd STAT PLOT; ON, Type (6th option; Data: L; Data Axis Y; Mark: “box” Sample Questions: 1. If x is a normally distributed variable with a mean of 30 and a standard deviation of 6, find the following probabilities: a. P ( x ! 30 ) b. P ( x ! 18 ) c. P ( x ! 36 ) d. P ( 24 ! x ! 39 ) 2. Determine the z scores that produce the following probabilities. a. 0.20 lies to the right of the z-score b. 0.40 lies to the right of the z-score c. 0.875 lies to the right of the z-score. d. 0.6328 lies between z and -z. 3. Consider the following data set: {1, 5, 2, 3, 4, 3, 3, 4, 4, 3, 2} to answer the following questions. a. Make a histogram for this data b. Make a Box-n-Whisker plot for the data. <----+----+----+----+----+----+----+----+----+----> c. What is the Pearson’s Index for the set of data? d. Make a normal quartile plot for the data e. Interpret the results from parts a through d. Homework: Pages 286 - 291 Exercises: 1, 5, 9, 13, 17, 21, 25, 29, 33, and 37 Exercises: 3, 7, 11, 15, 19, 23, 27, 31, 35, and 39 Exercises: 2, 6, 10, 14, 18, 22, 26, 30, 34, and 38 Exercises: 4, 8, 12, 16, 20, 24, 28, 32, 36, and 40