1. Chapter 3 The Normal Distributions

2. Chapter outline 1. Density curves 2. Normal distributions 3. The 68-95-99.7 rule 4. The standard normal distribution 5. Normal distribution calculations - 1: proportion? 6. Normal distribution calculations - 2: z- score?

3. Density curves A density curve is a curve that –1. is always on or above the x-axis –2. Has area exactly 1 underneath it. Special Case : Normal curve A density curve describes the overall pattern of a distribution. Areas under the density curve represent proportions of the total number of observations.

4. Density curves

7. Properties of density curve: –Median of a density curve: the equal-area point  the point that divides the area under the curve in half. –Mean of a density curve: the balance point, at which the curve would balance if made of solid material. Notation: mean ( ), standard deviation ( ), for a density curve.

8. Density curves

9. Normal distributions Possible values vary from Notation: A density curve - –It is single peaked and bell-shaped. –It never hits x-axis. It is above x-axis. –Centered at. That is, determines the location of center. –Having spread around the mean

10. Figure 3.7 (P.62) Two normal curves, showing the mean and standard deviation

11. The 68-95-99.7 rule For : 1. 68% of the observations fall within of 2. 95% of the observations fall within 2 of 3. 99.7% of the observations fall within 3 of

12. The 68-95-99.7 rule

13. Example 3.2 (P.63)

14. The standard normal distribution Mean=0, standard deviation =1 Notation: If x follows, follows

15. The standard normal distribution Example 3.3 (P.65) Example 3.4 (P.66)

16. How to use Table A To find a proportion: start with values on edges and find a value within the table To find a z-score: start in the middle of table and read the edges.

17. Normal distribution calculations 1: proportion? By using Table A: areas under the curve of N(0,1) are provided. –1. State in terms of –2. State the problem in terms of x –3. Standardize x in terms of z –4. Draw a picture to show the area we are interested in –5. Use Table A to find the required area Area to the left? Area to the right? Area in between?

18. Normal distribution calculations 1: proportion? Example 3.5 (P.68) Example 3.6 (P.69) Example 3.7 (P.70)

19. Normal distribution calculations 2: z-scores? So far, we find a proportion using specific value(s) on x-axis. Question: What if proportion is given and we want to find the specific value(s) on x-axis that give(s) given proportion? –1. State in terms of –2. State the problem in terms of z –3. Use Table A –4. Unstandardize from z to x (if needed)

20. Normal distribution calculations 1: proportion? Exercise 3.10 (P.70 ) Exercise 3.20 (P. 75)

21. Normal distribution calculations 2: z-scores? Example 3.8 (P.72) Exercise 3.12 (P.73)