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Understanding Normal Distributions: Density Curves, Rules, and Calculations

Learn about density curves, normal distributions, the 68-95-99.7 rule, and how to calculate proportions and z-scores using the standard normal distribution.

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Understanding Normal Distributions: Density Curves, Rules, and Calculations

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  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

  5. Density curves

  6. Density curves

  7. Density curves • 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. The 68-95-99.7 rule • 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)

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