1. Slide 13 - 1 Copyright © 2009 Pearson Education, Inc. Types of Distributions Rectangular Distribution J-shaped distribution

2. Slide 13 - 2 Copyright © 2009 Pearson Education, Inc. Types of Distributions continued Bimodal Skewed to right

3. Slide 13 - 3 Copyright © 2009 Pearson Education, Inc. Types of Distributions continued Skewed to left Normal

4. Slide 13 - 4 Copyright © 2009 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.

5. Slide 13 - 5 Copyright © 2009 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).

6. Slide 13 - 6 Copyright © 2009 Pearson Education, Inc. z-Scores z-scores determine how far, in terms of standard deviations, a given score is from the mean of the distribution.

7. Slide 13 - 7 Copyright © 2009 Pearson Education, Inc. 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) 55b) 60c) 43 a) A score of 55 is one standard deviation above the mean.

8. Slide 13 - 8 Copyright © 2009 Pearson Education, Inc. Example: z-scores continued b) A score of 60 is 2 standard deviations above the mean. c) A score of 43 is 1.4 standard deviations below the mean.

9. Slide 13 - 9 Copyright © 2009 Pearson Education, Inc. To Find 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 on page 387-388.

10. Slide 13 - 10 Copyright © 2009 Pearson Education, Inc. To Find the Percent of Data Between any Two Values continued 4. a) When finding the percent of data between two z- scores on opposite sides of the mean (when one z-score is positive and the other is negative), you find the sum of the individual percents. b) When finding the percent of data between two z- scores on the same side of the mean (when both z-scores are positive or both are negative), subtract the smaller percent from the larger percent.

11. Slide 13 - 11 Copyright © 2009 Pearson Education, Inc. To Find the Percent of Data Between any Two Values continued c) When finding the percent of data to the right of a positive z-score or to the left of a negative z-score, subtract the percent of data between 0 and z from 50%. d) When finding the percent of data to the left of a positive z-score or to the right of a negative z- score, add the percent of data between 0 and z to 50%.

12. Slide 13 - 12 Copyright © 2009 Pearson Education, Inc. 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.

13. Slide 13 - 13 Copyright © 2009 Pearson Education, Inc. 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.9772-.1587 =.8185=81.85%

14. Slide 13 - 14 Copyright © 2009 Pearson Education, Inc. Solution continued c. at least 17 mind. less than 8 min

15. Slide 13 - 15 Copyright © 2009 Pearson Education, Inc. Slide 13 - 15 Copyright © 2009 Pearson Education, Inc. 9.4 Linear Correlation and Regression

16. Slide 13 - 16 Copyright © 2009 Pearson Education, Inc. Linear Correlation Linear correlation is used to determine whether there is a relationship between two quantities and, if so, how strong the relationship is.

17. Slide 13 - 17 Copyright © 2009 Pearson Education, Inc. Linear Correlation – The linear correlation coefficient, r, is a unitless measure that describes the strength of the linear relationship between two variables. If the value is positive, as one variable increases, the other increases. If the value is negative, as one variable increases, the other decreases. The variable, r, will always be a value between –1 and 1 inclusive.

18. Slide 13 - 18 Copyright © 2009 Pearson Education, Inc. Scatter Diagrams A visual aid used with correlation is the scatter diagram, a plot of points (bivariate data). – The independent variable, x, generally is a quantity that can be controlled. – The dependent variable, y, is the other variable. The value of r is a measure of how far a set of points varies from a straight line. – The greater the spread, the weaker the correlation and the closer the r value is to 0. – The smaller the spread, the stronger the correlation and the closer the r value is to 1.

19. Slide 13 - 19 Copyright © 2009 Pearson Education, Inc. Correlation

20. Slide 13 - 20 Copyright © 2009 Pearson Education, Inc. Correlation

21. Slide 13 - 21 Copyright © 2009 Pearson Education, Inc. Linear Correlation Coefficient The formula to calculate the correlation coefficient (r) is as follows:

22. Slide 13 - 22 Copyright © 2009 Pearson Education, Inc. Example: Words Per Minute versus Mistakes There are five applicants applying for a job as a medical transcriptionist. The following shows the results of the applicants when asked to type a chart. Determine the correlation coefficient between the words per minute typed and the number of mistakes. 934Nancy 1041Kendra 1253Phillip 1167George 824Ellen MistakesWords per MinuteApplicant

23. Slide 13 - 23 Copyright © 2009 Pearson Education, Inc. Solution We will call the words typed per minute, x, and the mistakes, y. List the values of x and y and calculate the necessary sums. 306811156934  xy = 2,281  y 2 = 510  x 2 =10,711  y = 50  x = 219 10 12 11 8 y Mistakes xyy2y2 x2x2 x 41 53 67 24 WPM 4101001681 6361442809 7371214489 19264576

24. Slide 13 - 24 Copyright © 2009 Pearson Education, Inc. Solution continued The n in the formula represents the number of pieces of data. Here n = 5.

25. Slide 13 - 25 Copyright © 2009 Pearson Education, Inc. Solution continued

26. Slide 13 - 26 Copyright © 2009 Pearson Education, Inc. Solution continued Since 0.86 is fairly close to 1, there is a fairly strong positive correlation. This result implies that the more words typed per minute, the more mistakes made.

27. Slide 13 - 27 Copyright © 2009 Pearson Education, Inc. Linear Regression Linear regression is the process of determining the linear relationship between two variables. The line of best fit (regression line or the least squares line) is the line such that the sum of the squares of the vertical distances from the line to the data points (on a scatter diagram) is a minimum.

28. Slide 13 - 28 Copyright © 2009 Pearson Education, Inc. The Line of Best Fit Equation:

29. Slide 13 - 29 Copyright © 2009 Pearson Education, Inc. Example Use the data in the previous example to find the equation of the line that relates the number of words per minute and the number of mistakes made while typing a chart. Graph the equation of the line of best fit on a scatter diagram that illustrates the set of bivariate points.

30. Slide 13 - 30 Copyright © 2009 Pearson Education, Inc. Solution From the previous results, we know that

31. Slide 13 - 31 Copyright © 2009 Pearson Education, Inc. Solution Now we find the y-intercept, b. Therefore the line of best fit is y = 0.081x + 6.452

32. Slide 13 - 32 Copyright © 2009 Pearson Education, Inc. Solution continued To graph y = 0.081x + 6.452, plot at least two points and draw the graph. 8.88230 8.07220 7.26210 yx

33. Slide 13 - 33 Copyright © 2009 Pearson Education, Inc. Solution continued