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AND Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 1 Chapter 13 Statistics Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 2 WHAT YOU WILL LEARN • • • • Sampling techniques Misuses of statistics Frequency distributions Histograms, frequency polygons, stem-and-leaf displays • Mode, median, mean, and midrange • Percentiles and quartiles Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 3 WHAT YOU WILL LEARN • Range and standard deviation • z-scores and the normal distribution • Correlation and regression Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 4 Section 7 The Normal Curve Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 5 Types of Distributions Rectangular Distribution J-shaped distribution Rectangular Distribution Frequency Values Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 6 Types of Distributions (continued) Bimodal Copyright © 2009 Pearson Education, Inc. Skewed to right Chapter 13 Section 7 – Slide 7 Types of Distributions (continued) Skewed to left Copyright © 2009 Pearson Education, Inc. Normal Chapter 13 Section 7 – Slide 8 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. Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 9 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). Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 10 z-Scores z-scores determine how far, in terms of standard deviations, a given score is from the mean of the distribution. value of piece of data mean x z standard deviation Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 11 Example: z-scores A normal distribution has a mean of 50 and a standard deviation of 5. Find z-scores for the following values. a) 55 b) 60 c) 43 value of piece of data mean a) z standard deviation 55 50 5 z55 1 5 5 A score of 55 is one standard deviation above the mean. Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 12 Example: z-scores (continued) 60 50 10 b) z60 2 5 5 A score of 60 is 2 standard deviations above the mean. 43 50 7 c) z43 1.4 5 5 A score of 43 is 1.4 standard deviations below the mean. Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 13 To Find the Percent of Data Between any Two Values 1. 2. 3. Draw a diagram of the normal curve, indicating the area or percent to be determined. Use the formula to convert the given values to z-scores. Indicate these zscores on the diagram. Look up the percent that corresponds to each z-score in Table 13.7. Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 14 To Find the Percent of Data Between any Two Values (continued) 4. a) When finding the percent of data to the left of a negative z-score, use Table 13.7(a). b) When finding the percent of data to the left of a positive z-score, use Table 13.7(b). c) When finding the percent of data to the right of a z-score, subtract the percent of data to the left of that z-score from 100%. d) When finding the percent of data between two z-scores, subtract the smaller percent from the larger percent. Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 15 Example Assume that the waiting times for customers at a popular restaurant before being seated for lunch are normally distributed with a mean of 12 minutes and a standard deviation of 3 min. a) Find the percent of customers who wait for at least 12 minutes before being seated. b) Find the percent of customers who wait between 9 and 18 minutes before being seated. c) Find the percent of customers who wait at least 17 minutes before being seated. d) Find the percent of customers who wait less than 8 minutes before being seated. Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 16 Solution a. wait for at least 12 minutes Since 12 minutes is the mean, half, or 50% of customers wait at least 12 min before being seated. b. between 9 and 18 minutes 9 12 z9 1.00 3 18 12 z18 2.00 3 Use table 13.7 on pages 889-89 in the 8th edition. 97.7% - 15.9% = 81.8% Copyright © 2009 Pearson Education, Inc. Chapter 13 Section 7 – Slide 17 Solution (continued) c. at least 17 min 17 12 z17 1.67 3 Use table 13.7(b) page 889. 100% - 95.3% = 4.7% Thus, 4.7% of customers wait at least 17 minutes. Copyright © 2009 Pearson Education, Inc. d. less than 8 min 8 12 z8 1.33 3 Use table 13.7(a) page 889. Thus, 9.2% of customers wait less than 8 minutes. Chapter 13 Section 7 – Slide 18