1. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Do Now: Scott was one of 50 junior boys to take the PSAT at his school. He scored 64 on the Critical Reading test. This placed Scott at the 68th percentile within the group of boys. Looking at all 50 boys’ Critical Reading scores, the mean was 58.2 and the standard deviation was 9.4. Calculate and compare Scott’s z-score among the group of test takers

2. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Normal curves -The distributions they describe are called Normal distributions

3. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. DEFINITION: The 68–95–99.7 rule In the Normal distribution with mean μ and standard deviation σ: Approximately 68% of the observations fall within 1σ of the mean μ. Approximately 95% of the observations fall within 2σ of μ. Approximately 99.7% of the observations fall within 3σ of μ

4. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution.

5. Example 1: Using the 68–95–99.7 rule Suppose we know that a distribution is exactly Normal for the scores of a New York State 6 th grade vocabulary exam with mean μ = 6.84 and standard deviation σ = 1.55. (a) Sketch a Normal curve for this distribution of test scores. Label the points that are one, two, and three standard deviations from the mean. (b) What percent of the NYS vocabulary scores are less than 3.74? Show your work. (c) What percent of the scores are between 5.29 and 9.94? Show your work.

6. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Example 2: The distribution of heights of young women aged 18 to 24 is a normal distribution with a mean µ = 64.5 and σ = 2.5). (a) Sketch a Normal curve for the distribution of young women’s heights. Label the points one, two, and three standard deviations from the mean. (b) What percent of young women have heights greater than 67 inches? (c) What percent of young women have heights between 62 and 72 inches?

7. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Example 3: Suppose we know that the average (μ) high school relationship is 42 days with a standard deviation (σ) of 2 days. (a)Assuming that the distribution of days is approximately Normal, make an accurate sketch of the distribution with the horizontal axis marked in days. (a)Between which standard deviations does the interval from 38-42 lie? (b)What percentage represents this interval? (c)What does this percentage represent in the context of this problem?

8. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Do Now: Suppose we know that the time it takes for college students to commute to their classroom from their dorms is normally distributed with an average time of μ = 8 minutes and a standard deviation of σ = 2 minutes. (a)Sketch a Normal curve for the distribution of college students commute time in minutes. (b)Determine the interval that represents the middle 68% of the data. (a)Interpret the values and percentage less than -1σ.

9. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Let’s look again at the Normal curve for the distribution in the Do Now: What if we wanted to find the percentage of values less than -1.21σ? Can we find this using our Normal curve?

10. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. The standard Normal table (z-table) **The z-table is a table of areas under the standard Normal curve. The table entry for each value z is the area under the curve to the left of z.

11. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Example: (a)Find the proportion of observations less than -1.25 (b)Find the percent of observations more than 0.81 (c)Find the percent of observations between -1.25 and 0.81 (d)Find the z value of all observations less than 34%

12. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. 1. Use the z-table to find the proportion of observations from a standard Normal distribution that fall in each of the following regions. In each case, sketch a standard Normal curve and shade the area representing the region. (a) z < 1.39 (b) z > −2.15 (c) −0.56 < z < 1.81 2. Use the z-table to find the value z from the standard Normal distribution that satisfies each of the following conditions. (a) The 20th percentile (b) 45% of all observations are greater than z

13. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution.

14. Exit Ticket A study of elite distance runners found a mean body weight of 63.1 kilograms (kg), with a standard deviation of 4.8 kg. (a) Assuming that the distribution of weights is approximately Normal, make an accurate sketch of the weight distribution with the horizontal axis marked in kilograms. (b) Use the 68–95–99.7 rule to find the proportion of runners whose body weight is between 48.7 and 67.9 kg. (c) What proportion of runners have body weights below 60 kg? (d) What proportion of runners have body weights above 70 kg?

15. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Do Now: (a)Find the proportion of observations less than -1.09 (b)Find the percent of observations more than 2.41 (c)Find the percent of observations between 0.92 and 2.11 (d)Find the z value of all observations more than 81%

16. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. How to Solve Problems Involving Normal Distributions 1.State the information you know. 2.Sketch your distribution and the area you are looking for. 3.Standardize your value in terms of a Normal standard variable (find your z-value) and use the z-table 4.Write your conclusion in the context of the problem.

17. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Example 1: On the driving range, Tiger Woods practices his swing with a particular club by hitting many, many balls. When Tiger hits his driver, the distance the ball travels follows a Normal distribution with mean 304 yards and standard deviation 8 yards. What percent of Tiger’s drives travel at least 290 yards?

18. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. Example 1(b): What distance would a ball have to travel to be at the 80th percentile of Tiger Woods’s drive lengths?

19. SWBAT: Describe and Apply the 68-95-99.7 Rule and the standard Normal Distribution. YOU TRY!! High levels of cholesterol in the blood increase the risk of heart disease. For 14-year-old boys, the distribution of blood cholesterol is approximately Normal with mean μ = 170 milligrams of cholesterol per deciliter of blood (mg/dl) and standard deviation σ = 30 mg/dl. (a) People with cholesterol levels between 200 and 240 mg/dl are at considerable risk for heart disease. What percent of 14-year-old boys have blood cholesterol between 200 and 240 mg/dl? (b) What is the first quartile of the distribution of blood cholesterol?