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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.

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Presentation on theme: "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."— Presentation transcript:

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

2  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 Normal Distribution

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

4 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 How many standard deviations is Jasmine above the mean? Z-Scores

5 Finding the Area or Probability

6 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 = ? Finding the Area or Probability

7  Using the Calculator:  2 nd  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

8  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 or about 3.6%  CONTEXT: Thus, approximately 3.6% of SAT scores are higher than 680. Finding the Area or Probability

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

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

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

12 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°Cbetween -2.43° and 0° more than 1.27°between 1.20° and 2.30° Normal Model Example

13 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? Normal Model Example

14  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 Day 2

15 Using the Table:  Look in the table for an area of  The exact area is not there, but 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

16 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

17 Using the Calculator:  2 nd  DISTR  invNorm(  Specify the desired percentile  invNorm(.25) = approximately  Thus the z-score is  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 90 th percentile. So you would use invNorm(.90). From Percentiles to Z-Scores

18 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

19 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? From Percentiles to Z-Scores

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


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