Transcript Chapter 5 5.1 * 5.2: The Standard Normal Distribution

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
Between z = 0 and z = 1.54
To the left of z = -1.28 or to the right of z = 1.28
To the left of z = 1.36
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
 Day 2: P. 245 # 5 – 17 ; 25, 26