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Math Tech IIII, Dec 7
Review of the Normal Distribution
Book Sections: 5.1, 5.2, & 5.3Essential Questions: What do I need to know about the normal
distribution in order to pass the unit 8 test?
Standards: PS.SPID.4, PS.SPMJ.1
Things to Know for This Test
1. The properties of the normal distribution
2. How to graph a normal curve
3. The relationships between the curve and the mean and standard
deviation
4. The standard normal distribution
5. How to compute any normal probability
6. How to go between a percent, probability and percentile
7. The reverse look-up problem (or how to find a corresponding
value from a probability)
The Test
• 20 total questions
Problems of every variety
Multiple choice
True-False
Graphing
Properties of the Normal Distribution
• A normal distribution with mean and standard deviation has
the following properties:
1. The mean, median, and mode are equal, and are located in the
middle of the curve.
2. The normal curve is bell-shaped and is symmetric about the mean.
3. The total area under the normal curve is equal to 1.
4. The normal curve approaches, but never touches, the x-axis.
5. Between μ – σ and μ + σ (in the center of the curve), the graph
curves downward. The graph curves upward to the left of μ – σ
and to the right of μ + σ. The point at which the curve changes are
called inflection points. This is what causes the ‘bell shape.’
Properties of the Normal Distribution
What YOU Need to Know about Standard Normal
• It is a unique normal distribution, there is only one of them.
• It is simple, it has a mean of 0 and a standard deviation of 1.
• Any normal distribution can become standard normal via a z-
score.
• It made the computation of normal probability a very doable
event with minimum use of technology (namely none).
Fun Facts of the Day
• The probability of something happening in a normal
distribution rounded to 2 decimal places is the percentile
(and can be called a probability).
• A standard deviation cannot be a negative number.
Graphing the Normal Distribution
• We are graphing this function for a given value of
the mean and standard deviation.
• To graph, select the purple Y= key, top left.
The function we are going to graph is the normalpdf
function (1st one in distributions) Its arguments are the
variable x, then mu , then sigma
We must then set the window to see the graph
Example
Graph the following normal
distributions: mean = 2, SD = .5
mean = 2, SD = 1
Window
Xmin = 0
Xmax = 4
Xscl = .5
Ymin = 0
Ymax = 1
Yscl = .1
Example 1
• In the following example:
Example 2
• Consider the curves
shown at the right.
Which has the
greatest mean and
standard deviation?
Estimate the mean, μ
of each.
How The Calculator Works
• The calculator can compute any normal
probability1. Use the distribution values (four arguments)
Either way, you are using the normalcdf
distribution
[2nd][Vars] select 2 gets you this function
The Normal Distribution in Texas
1. To use the graphing calculator to compute probabilities (or areas)
using the normal distribution.
Use the normalcdf function (2nd)(Vars)(2)
• This function can be used in many ways:
With any distribution, using 4 arguments; LB, UB, μ, σ
Probability Models
• P(x < a) = area left of a (probability model a)
Calculator Computation: normalcdf(-1000, a, μ, σ
The words: less than or fewer than
• P(x > b) = area right of b (probability model b)
Calculator Computation : normalcdf(b, 1000, μ, σ
The words: more than or greater than
• P(c < x < d) = the area between c and d (probability model c)
Calculator Computation : normalcdf(c, d, μ, σ
The words: between
• invNorm(p(x), μ, σ
• The result is a random variable value in the
distribution, not a probability
The Reverse Lookup
• The words – what percent – implies a probability
computation, not a reverse lookup. This asks for a
percent – compute a probability then convert it to
a percent. How? Round to 2 decimal places then multiply by 100.
• In a reverse lookup, you are provided with a
percentile, which is converted to a p(x).
Caution
Example 3
The car speeds that Trooper Tracy clocks along a stretch of I-85 are
normally distributed with a mean of 67 mph and a standard
deviation of 9 mph. Compute each probability:
a) That the next car’s speed will be more that 80.
b) That the next car’s speed will be between 60 and 75.
c) That the next car’s speed will be less than 55.
Example 4
The car speeds that Trooper Tracy clocks along a stretch of I-85 are
normally distributed with a mean of 67 mph and a standard
deviation of 9 mph. What speed corresponds to the 60th percentile?
Example 5
The car speeds that Trooper Tracy clocks along a stretch of I-85 are
normally distributed with a mean of 67 mph and a standard
deviation of 9 mph. Tracy only gives tickets to cars in the 85th
percentile. At what speed does she stop someone?
Classwork: CW 12/7/16, 1-24
Homework – None
15)
Properties of the Normal Distribution
• A normal distribution with mean and standard deviation has
the following properties:
1. The mean, median, and mode are equal, and are located in the
middle of the curve.
2. The normal curve is bell-shaped and is symmetric about the mean.
3. The total area under the normal curve is equal to 1.
4. The normal curve approaches, but never touches, the x-axis.
5. Between μ – σ and μ + σ (in the center of the curve), the graph
curves downward. The graph curves upward to the left of μ – σ
and to the right of μ + σ. The point at which the curve changes are
called inflection points. This is what causes the ‘bell shape.’
26)