1. Copyright © 2012 by Nelson Education Limited. Chapter 4 The Normal Curve 4-1

2. Copyright © 2012 by Nelson Education Limited. The Normal Curve Z scores The use of the Normal Curve table (Appendix A) Finding areas above and below Z scores Finding probabilities In this presentation you will learn about: 4-2

3. Copyright © 2012 by Nelson Education Limited. Bell Shaped Unimodal Symmetrical Unskewed Mode, Median, and Mean are same value Theoretical Normal Curve 4-3

4. Copyright © 2012 by Nelson Education Limited. Distances on horizontal axis always cut off the same area. We can use this property to describe areas above or below any point. Theoretical Normal Curve: Specific Areas 4-4

5. Copyright © 2012 by Nelson Education Limited. To find areas, first compute a Z score. The formula for computing a Z score is * This formula changes a “raw” score (X i ) to a standard deviation or Z score. second, use Appendix A to find the area above or below a Z score. *Converting original scores in a population is done using the same method. Using the Normal Curve: Z Scores 4-5

6. Copyright © 2012 by Nelson Education Limited. Using the Normal Curve: Appendix A 4-6

7. Copyright © 2012 by Nelson Education Limited. Appendix A has three columns. –(a) = Z score –(b) = areas between the mean and the Z score Using the Normal Curve: Appendix A (continued) 4-7

8. Copyright © 2012 by Nelson Education Limited. –( c) = areas beyond the Z score Using the Normal Curve: Appendix A (continued) 4-8

9. Copyright © 2012 by Nelson Education Limited. The normal curve table can be used to find the: 1. area between a Z score and the mean. (Section 5.3) 2. area either above or below a Z score (5.4) * 3. area between two Z scores (5.5) 4. probability of randomly selected score (5.6) * * Only these are demonstrated in this presentation Using Appendix A to Describe Areas Under the Normal Curve 4-9

10. Copyright © 2012 by Nelson Education Limited. Find your Z score in column (a). To find area below a positive score: –Add column (b) area to 0.50. To find area above a positive score –Look in column (c). (a)(b)(c)... 1.660.45150.0485 1.670.45250.0475 1.680.45350.0465... How to Find Area Above or Below a (Positive) Z Score 4-10

11. Copyright © 2012 by Nelson Education Limited. A person has a height of 73 inches in a distribution of height where, = 68 inches and s = 3 inches. The person’s score as a Z score is: How to Find Area Below a (Positive) Z Score: An Example 4-11

12. Copyright © 2012 by Nelson Education Limited. To find the area below a positive Z score, we consult the normal curve table (Appendix A) to find the area between the score and the mean (column b): 0.4525. Then we add this area to the area below the mean: 0.5000, or 0.4525 + 0.5000 = 0.9525. Areas can be expressed as percentages: 95.25%. The area below a Z score of +1.67 is 95.25%. A person with a height of 73 inches is taller than 95.25% of all persons. How to Find Area Below a (Positive) Z Score: An Example (continued) Normal curve with Z=+1.67 4-12

13. Copyright © 2012 by Nelson Education Limited. Find your Z score in column (a). To find area below a negative score: –Look in column (c). To find area above a negative score –Add column (b) area to 0.50 (a)(b)(c)... 1.660.45150.0485 1.670.45250.0475 1.680.45350.0465... How to Find Area Above or Below a (Negative) Z Score 4-13

14. Copyright © 2012 by Nelson Education Limited. On the other hand, the Z score for a person with a height of 63 is: -1.67. To find the area below a negative score we use column c in Appendix A: the area below a Z score of -1.67 is 0.0475, or 4.75%. This person is taller than 4.75% of all persons. How to Find Area Below a (Negative) Z Score: An Example Normal curve with Z=-1.67 4-14

15. Copyright © 2012 by Nelson Education Limited. Summary: Finding an Area Above or Below a Z Score 4-15

16. Copyright © 2012 by Nelson Education Limited. Areas under the curve can also be expressed as probabilities. Probabilities are proportions and range from 0.00 to 1.00. The higher the value, the greater the probability (the more likely the event). Probability is essential for understanding inferential statistics in Part II of text. Finding Probabilities 4-16

17. Copyright © 2012 by Nelson Education Limited. If a distribution has: =13 and s = 4, what is the probability of randomly selecting a score of 19 or more? 1. Use the formula for computing a Z score: For X i = 19, Z = 1.50 2. Find area above in column (c). 3. Probability is 0.0668 of randomly selecting a score of 19 or more. (a)(b)(c)... 1.490.43190.0681 1.500.43320.0668 1.510.43450.0655... Finding Probabilities: An Example 4-17