review of the normal distribution book sections: 5.1, 5.2, & 5 · 1. the mean, median, and mode...

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.’

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



deviation of 9 mph. What speed corresponds to the 60th percentile?

Example 5



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


• 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)

review of the normal distribution book sections: 5.1, 5.2, & 5 · 1. the mean, median, and mode...

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