1. Normal Distributions Section 2.1.2

2. Starter A density curve starts at the origin and follows the line y = 2x. At some point on the line where x = p, the curve drops vertically to return to the x axis. 1.Draw the curve. 2.What is the value of p? 3.What is the median of the curve? –Show it as a vertical line on the curve.

3. Answer

4. Today’s Objectives Draw a normal curve and show μ, µ±σ, µ±2σ, µ±3σ on the graph Use the Empirical Rule (a.k.a. 68-95 Rule) to answer questions about percents and percentiles

5. Normal Curves Draw a bell-shaped curve above an x axis Draw the vertical line of symmetry –Label the x axis “μ” at this point Show the two “inflection points” on the curve The left inflection point is where the curve stops getting more steep and starts getting less steep The other is symmetric to it about the mean line –Label the x axis “µ+σ” and “µ-σ” below the points Using the same scale, label the x axis with µ plus and minus 2σ and 3σ

6. Normal Curves Normal curves are special case density curves –The area under the curve is 1 This is true of ALL density curves –The curve is symmetric and “bell-shaped” So mean = median We normally speak of the mean rather than median –The inflection points of the curve are one standard deviation (σ) above and below the mean

7. The Empirical Rule or: 68-95 Rule In a normal distribution with mean μ and standard deviation σ: –This is called the N(μ, σ) distribution –About 68% of the observations fall within σ of μ –About 95% fall within 2σ of μ –About 99.7% fall within 3σ of μ

8. Example Suppose the heights of American men are known to be N(69 in, 2.5 in) –Draw the normal curve and label the axis –What percent of men are between 69 inches and 71.5 inches tall? Since 68% are between 66.5 and 71.5, and the graph is symmetric, there are 34% between 69 and 71.5 inches tall.

9. Example Continued What percent of men are taller than 74 in? –Since 95% of observations fall within ±2σ of μ, then 5% fall outside those borders. –By symmetry, 2.5% fall more than 2σ above the mean. In this case, that is 69 + 2 x 2.5 = 74 in So 2.5% of men are taller than 74 in

10. Example Concluded In what percentile is a man who is 71.5 in tall? Recall that “percentile” means the percent of observations equal to or less than the specified value –By definition, 50% fall below 69 inches –71.5 inches is one σ above the mean, so 34% must fall between 69 and 71.5 –Thus 71.5 inches is the 84 th percentile 50 + 34 = 84

11. Exploring Normal Data 50 Fathoms Demo 3 –What Do Normal Data Look Like?

12. Today’s Objectives Draw a normal curve and show μ, µ±σ, µ±2σ, µ±3σ on the graph Use the Empirical Rule (a.k.a. 68-95 Rule) to answer questions about percents and percentiles

13. Homework Read pages 73 – 77 Do problems 6 – 9