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Section 13.6 The Normal Curve Copyright 2013, 2010, 2007, Pearson, Education, Inc. INB Table of Contents Date 2.3-2 Topic Page # July 14, 2014 Section 13.6 Examples 68 July 14, 2014 Section 13.6 Notes 69 July 14, 2014 Negative z-chart 70 July 14, 2014 Positive z-chart 71 July 14, 2014 Test 4 Practice Test 72 July 14, 2014 Test 4 Practice Test Workspace 73 Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Rectangular Distribution J-shaped Distribution Bimodal Distribution Skewed Distribution Normal Distribution z-Scores 13.6-3 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Rectangular Distribution All the observed values occur with the same frequency. 13.6-4 Copyright 2013, 2010, 2007, Pearson, Education, Inc. J-shaped Distribution The frequency is either constantly increasing or constantly decreasing. 13.6-5 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Bimodal Distribution Two nonadjacent values occur more frequently than any other values in a set of data. 13.6-6 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Skewed Distribution Has more of a “tail” on one side than the other. 13.6-7 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Skewed Distribution Smoothing the histograms of the skewed distributions to form curves. 13.6-8 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Skewed Distribution The relationship between the mean, median, and mode for curves that are skewed to the right and left. 13.6-9 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Normal Distribution The most important distribution is the normal distribution. 13.6-10 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Properties of a Normal Distribution The graph of a normal distribution is called the normal curve. The normal curve is bell shaped and symmetric about the mean. In a normal distribution, the mean, median, and mode all have the same value and all occur at the center of the distribution. 13.6-11 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Empirical Rule Approximately 68% of all the data lie within one standard deviation of the mean (in both directions). Approximately 95% of all the data lie within two standard deviations of the mean (in both directions). Approximately 99.7% of all the data lie within three standard deviations of the mean (in both directions). 13.6-12 Copyright 2013, 2010, 2007, Pearson, Education, Inc. z-Scores z-scores (or standard scores) determine how far, in terms of standard deviations, a given score is from the mean of the distribution. 13.6-13 Copyright 2013, 2010, 2007, Pearson, Education, Inc. z-Scores The formula for finding z-scores (or standard scores) is value of piece of data mean z standard deviation x 13.6-14 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Finding z-scores A normal distribution has a mean of 80 and a standard deviation of 10. Find z-scores for the following values. a) 90 b) 95 c) 80 d) 64 13.6-15 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Finding z-scores a) 90 13.6-16 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Finding z-scores b) 95 13.6-17 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Finding z-scores c) 80 13.6-18 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Finding z-scores d) 64 13.6-19 Copyright 2013, 2010, 2007, Pearson, Education, Inc. To Determine the Percent of Data Between any Two Values 1. Draw a diagram of the normal curve indicating the area or percent to be determined. 2. Use the formula to convert the given values to z-scores. Indicate these z-scores on the diagram. 3. Look up the percent that corresponds to each z-score in Table 13.7. 13.6-20 Copyright 2013, 2010, 2007, Pearson, Education, Inc. To Determine the Percent of Data Between any Two Values a) When finding the percent of data to the left of a negative z-score, use Table 13.7(a). 13.6-21 Copyright 2013, 2010, 2007, Pearson, Education, Inc. To Determine the Percent of Data Between any Two Values b) When finding the percent of data to the left of a positive z-score, use Table 13.7(b). 13.6-22 Copyright 2013, 2010, 2007, Pearson, Education, Inc. To Determine the Percent of Data Between any Two Values c) When finding the percent of data to the right of a z-score, subtract the percent of data to the left of that zscore from 100%. 13.6-23 Copyright 2013, 2010, 2007, Pearson, Education, Inc. To Determine the Percent of Data Between any Two Values c) Or use the symmetry of a normal distribution. 13.6-24 Copyright 2013, 2010, 2007, Pearson, Education, Inc. To Determine the Percent of Data Between any Two Values d) When finding the percent of data between two z-scores, subtract the smaller percent from the larger percent. 13.6-25 Copyright 2013, 2010, 2007, Pearson, Education, Inc. To Determine the Percent of Data Between any Two Values 4. Change the areas you found in Step 3 to percents as explained earlier. 13.6-26 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Horseback Rides Assume that the length of time for a horseback ride on the trail at Triple R Ranch is normally distributed with a mean of 3.2 hours and a standard deviation of 0.4 hour. a) What percent of horseback rides last at least 3.2 hours? 13.6-27 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Horseback Rides b) What percent of horseback rides last less than 2.8 hours? 13.6-28 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Horseback Rides c) What percent of horseback rides are at least 3.7 hours? 13.6-29 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Horseback Rides d) What percent of horseback rides are between 2.8 hours and 4.0 hours? 13.6-30 Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Horseback Rides e) In a random sample of 500 horseback rides at Triple R Ranch, how many are at least 3.7 hours? 13.6-32 Copyright 2013, 2010, 2007, Pearson, Education, Inc.