Chapter 5 5.1 – 5.4: The Normal Model

1. CHS Statistics Chapter 55.1 – 5.4: The Normal Model Objective: To apply prior knowledge of the Normal model and understand the concepts of positions on the Normal model

2. The Normal Distribution If a continuous random variable has a distribution with a graph that is symmetric and bell-shaped has a Normal distribution. Total area under curve is 1 –why? Recall: To apply the Normal distribution characteristics, you must be under the assumption that the data would follow a Normal distribution (i.e. be fairly symmetric). Otherwise, the Normal model does not apply!

3. The Standard Normal Distribution A Standard Normal Distribution is a Normal probability distribution that has a mean of zero (0) and a standard deviation of one (1).

4. Z-Scores Recall: Z-scores tell us a value’s distance from the mean in terms of standard deviations. Formula: Example: On the SAT, Jasmine earned a 680 on her math portion. Suppose the mean SAT math score is 620 with a standard deviation of 33.33. How many standard deviations is Jasmine above the mean?

5. Finding the Area or Probability • Sometimes a z-score will not be exactly 1, 2, 3 • Using the Table: • The figure shows us how to find the area to the left when we have a z-score of 1.80:

6. Finding the Area or Probability Practice using the tables: • What is the area or probability to the left of a z-score of 1.15? • What is the area or probability to the left of a z-score of – 0.24? • To the right of z = - 0.24?

7. Finding the Area or Probability • Using the Calculator: • 2nd DISTR  • Normalpdf( calculates the x-values for graphing a normal curve. You probably won’t us this very often. • Normalcdf( finds the proportion of area under the curve between two z-score cut points by specifying Normalcdf( Lower bound, Upper bound) • Sometimes the left and right z-scores will be given to you, as you would want to find the percentage between. However, that is not always the case…

8. Finding the Area or Probability • In the last example we found that 680 has a z-score of 1.8, thus is 1.8 standard deviations away from the mean. • The z-score is 1.8, so that is the left cut point. • Theoretically the standard Normal model extends rightward forever, but you can’t tell the calculator to use infinity as the right cut point. It is suggested that you use 99 (or -99) when you want to use infinity as your cut point. • Normalcdf(1.8,99) = approx. 0.0359 or about 3.6% • CONTEXT: Thus, approximately 3.6% of SAT scores are higher than 680.

9. Finding the Area or Probability Practice using the calculator: • Find the area to the left of z = 1.23. • Find the area to the right of z = 1.23 • Find the area between z = 1.23 and z = -0.75

10. Finding the Area or Probability Using any method, find the area: To the left of z = -0.99 To the right of z = -0.65 To the left of z = -2.57 To the right of z = 1.06

11. Finding the Area or Probability Using any method, find the area: In between z = -1.5 and z = 1.25 To the left of z = 1.36 Between z = 0 and z = 1.54 To the left of z = -1.28 or to the right of z = 1.28

12. Normal Model Example A thermometers company is supposed to give readings of 0° C at the freezing point of waters. Tests on a large sample reveal that at the freezing point some give readings below 0° and some above 0°. Assume that the mean is 0°C with a standard deviation of 1°C. Also assume the readings are normally distributed. If one thermometer is randomly selected, find the probability that, at the freezing point of water, the reading is Between 0° and +1.58°C between -2.43° and 0° more than 1.27° between 1.20° and 2.30°

13. Normal Model Example Suppose a Normal model describes the fuel efficiency of cars currently registered in your state. The mean is 24 mpg, with a standard deviation of 6 mpg. Provide sketches for each solution. • What percent of all cars get less than 15 mpg? • What percent of all cars get between 20 and 30 mpg? • What percent of cars get more than 40 mpg?

14. From Percentiles to Z-Scores Day 2 Sometimes we start with areas and need to find the corresponding z-score or even the original data value. Example: What z-score represents the first quartile in a Normal model?

15. From Percentiles to Z-Scores Using the Table: Look in the table for an area of 0.2500. The exact area is not there, but 0.2514 is pretty close. This figure is associated with z = –0.67, so the first quartile is 0.67 standard deviations below the mean.

16. From Percentiles to Z-Scores Using the Table: The area to the left of a z-score is 89%. What is that z-score? The area to the right of a z-score is 52%. What is that z-score?

17. From Percentiles to Z-Scores Using the Calculator: • 2nd DISTR  invNorm( • Specify the desired percentile • invNorm(.25) = approximately -0.674 • Thus the z-score is -0.674 • 0.674 standard deviations below the mean • Be careful with percentiles: If you are asked what z-score cuts off the highest 10% of a Normal model remember that is the 90th percentile. So you would use invNorm(.90).

18. From Percentiles to Z-Scores Let’s try the same examples as before using the calculator: The area to the left of a z-score is 89%. What is that z-score? The area to the right of a z-score is 52%. What is that z-score?

19. From Percentiles to Z-Scores Suppose a Normal model describes the fuel efficiency of cars currently registered in your state. The mean is 24 mpg, with a standard deviation of 6 mpg. Provide sketches for each solution. • Describe the fuel efficiency of the worst 20% of all cars. • What gas mileage represents the third quartile? • Describe the gas mileage of the most efficient 5% of all cars. • What gas mileage would you consider unusual? Why?

20. Assignment Day 1: P. 239 # 10 – 32 Even, 33 – 35 Day 2: P. 245 # 5 – 17 ; 25, 26