2. Copyright © 2011 Pearson Education, Inc. Putting Statistics to Work

3. Copyright © 2011 Pearson Education, Inc. Slide 6-3 Unit 6C The Normal Distribution

4. 6-C Copyright © 2011 Pearson Education, Inc. Slide 6-4 The Normal Distribution The normal distribution is a symmetric, bell- shaped distribution with a single peak. Its peak corresponds to the mean, median, and mode of the distribution.

5. 6-C Copyright © 2011 Pearson Education, Inc. Slide 6-5 1.Most data values are clustered near the mean, giving the distribution a well-defined single peak. 2.Data values are spread evenly around the mean, making the distribution symmetric. 3.Larger deviations from the mean are increasingly rare, producing the tapering tails of the distribution. 4.Individual data values result from a combination of many different factors. A data set satisfying the following criteria is likely to have a nearly normal distribution. Conditions for a Normal Distribution

6. 6-C Copyright © 2011 Pearson Education, Inc. Slide 6-6 The 68-95-99.7 Rule for a Normal Distribution

7. 6-C Applying the 68-95-99.7 Rule The resting heart rates for a sample of people are normally distributed with a mean of 70 and a standard deviation of 15. Find the percentage of heart rates that are Less than 55 Less than 40 Less than 100 Between 55 and 85 Greater than 115 Copyright © 2011 Pearson Education, Inc. Slide 6-7

8. 6-C Copyright © 2011 Pearson Education, Inc. Slide 6-8 Applying the 68-95-99.7 Rule for the Sample of Resting Heart Rates 7085100115554025 34% 47.5%

9. 6-C Copyright © 2011 Pearson Education, Inc. Slide 6-9 Standard Scores The number of standard deviations that a data value lies above or below the mean is called its standard score (or z-score), defined by Data Value above the mean below the mean Standard Score positive negative → → What is the z-score for the mean?

10. 6-C Copyright © 2011 Pearson Education, Inc. Slide 6-10 Standard Scores Example: If the mean were 21 with a standard deviation of 4.7 for scores on a nationwide test, find the z-score for a 30. What does this mean? This means that a test score of 30 would be about 1.91 standard deviations above the mean of 21.

11. 6-C Copyright © 2011 Pearson Education, Inc. Slide 6-11 The nth percentile of a data set is the smallest value in the set with the property that n% of the data are less than or equal to it. A data value that lies between two percentiles is said to lie in the lower percentile. Standard Scores and Percentiles

12. 6-C Copyright © 2011 Pearson Education, Inc. Slide 6-12 Standard Scores and Percentiles

13. 6-C Application of Standard Scores and Percentiles Scores on a chemistry exam were normally distributed with a mean of 67 and a standard deviation of 8 What is the standard score for an exam score of 67? Find the percentile for that exam score. What is the standard score for an exam score of 88? Find the percentile for that exam score. What is the standard score for an exam score of 59? Find the percentile for that exam score. Copyright © 2011 Pearson Education, Inc. Slide 6-13