Chapter 5 5.1 – 5.4: The Normal Model CHS Statistics Chapter 5 5.1 – 5.4: The Normal Model Objective: To apply prior knowledge of the Normal model and understand the concepts of positions on the Normal model

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!

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).

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?

Finding the Area or Probability Sometimes a z-score will not be exactly ±1, ±2, or ±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:

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?

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…

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.

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

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

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

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°

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?

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?

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.

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?

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).

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?

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?

Assignment Day 1: P. 239 # 10 – 32 Even, 33 – 35