1. 1 From density curve to normal distribution curve (normal curve, bell curve) Class 18

2. 2 Class Objective After this class, you will be able to Use the 68-95-99.7 rule of the Normal Distribution Curve (Normal Curve/Bell Curve) to describe a population

3. Homework Check Assignment: – Density Worksheet 3

4. Example Population: 180 students at TRMC Quantitative Variable: Weight in Pounds Frequency Table: Based on the above Frequency Table, construct a histogram 4 Lower LimitUpper LimitFrequency 119.5129.520 129.5139.540 139.5149.560 149.5159.540 159.5169.520

5. Histogram 5

6. From Histogram to Normal Distribution Curve 6

7. Population = 180 students Population Mean (weight in pounds) = 144.5 7

8. Population = 180 students Population Mean (weight in pounds) = 144.5 8 Interpretation: 90 Students / 50 % of the population weighs more than 144.5 pounds

9. Now you try it – Write the interpretation 9

10. Normal Curve / Bell Curve Symmetric Single-peaked Bell Shaped – inflection points between cupping upward and downward The curve approaches the horizontal axis but never touches or crosses it 10

11. Standard deviation and mean of a normal curve Mean fixes the center of the curve – Changing mean does not change the shape of the curve Standard deviation fixed the shape – Changing standard deviation changes the shape of the curve. 11

12. Construct a Normal Curve in terms of the mean and standard deviation 12

13. 13 Interpreting the Standard Deviation for Bell-Shaped Curves: The Empirical Rule For any bell-shaped curve, approximately 68% of the values fall within 1 standard deviation of the mean in either direction 95% of the values fall within 2 standard deviations of the mean in either direction 99.7% of the values fall within 3 standard deviations of the mean in either direction Note: ~0.3% fall farther than 3 standard deviations from mean

14. 14 Example 2.19 Women’s Heights revisited Mean height for the 199 British women is 1602 mm and standard deviation is 62.4 mm. 68% of the 199 heights would fall in the range 1602  62.4, or 1539.6 to 1664.4 mm 95% of the heights would fall in the interval 1602  2(62.4), or 1477.2 to 1726.8 mm 99.7% of the heights would fall in the interval 1602  3(62.4), or 1414.8 to 1789.2 mm

15. 15 Example 2.19 Women’s Heights revisited Note: Not perfect, but follows Empirical Rule quite well

16. The 68-95-99.7 Rule of a Normal Curve (Normal Distribution) 68% of the observation fall within one standard deviation of the mean 95% of the observation fall within two standard deviation of the mean 99.7% of the observation fall with three standard deviations of the mean 16

17. Example Population: American Young Women Quantitative Variable: Height in inches The height distribution is normal. Mean = 65 inches Standard Deviation = 2.5 inches Question: – Based on these information and the 68-95-99.7 rule of a normal distribution, what can you tell about the height distribution of the American young women? 17

18. The distribution of the height of the American Young Women 18

19. Height distribution of Young Women in America 34% of young women are between 65 inches and 67.5 inches tall 95% of young women’s height s are between 60 inches and 70 inches The shortest 2.5% of young women are less than 60 inches tall 19

20. Now, you try it Population: American Young Men Quantitative Variable: Height in inches The height distribution is normal. Mean = 70 inches Standard Deviation = 2.5 inches Question: – Based on these information and the 68-95-99.7 rule of a normal distribution, what can you tell about the height distribution of the American young men? 20

21. Homework Assignment: – Chapter 2 – Exercise 2.98 and 2.103 Reading: – Chapter 2 – p. 49-50 21