**The Normal Distributions**

2.1

2.2 Standard Normal Calculations Vocabulary

Standard normal distribution

- A normal distribution with a mean of 0 and a standard deviation of 1

68-95-99.7 (

Empirical Rule

)

- given a density curve is normal, then the following is true:

within plus or minus one standard deviation is

68%

of data

withing plus or minus two standard deviation is

95%

of data

withing plus or minus three standard deviation is

99.7%

of data

Inverse Normal

- special family of bell-shaped, symmetric density curves that follow a complex formula

Standard Normal Distribution

- a normal distribution with a mean of 0 and a standard deviation of 1

VOCABULARY!!!!!!!!!!

Presented By Kayla F. Tashionna J. & Katraya J.

TODAY WE WILL...

Explain mathematical models

Learn what a density curve is

Find the mean and the median of a density curve.

find the z-score of an equation

BUT FIRST!!!!!!

Density Curve- the curve that represents proportions of the observations; and describes the overall pattern

Mathematical Model- an idealized representation

Median of a Density Curve- the "equal-areas point'' and denoted by M or MED

Mean of a Density Curve- the "balance point" and denoted by μ

Normal Curve- a special symmetric, mound shaped density curve with special characteristics

Standard Deviation of a Density Curve - denoted by a sigma

Standardized value- a z-score

Standardizing- converting data from original values to standard deviation units.

Standardized Value

Also known as the Z-Score

You can only find the z-score if the mean and the standard deviation is known...

x-mean

Z= ---------------

Standard deviation

EXAMPLE!

The weight of chocolate bars from a particular chocolate factory has a mean of 8 ounces with standard deviation of .1 ounce. What is the z-score corresponding to a weight of 8.17 ounces?

MEAN?

Standard deviation?

REMEMBER!

Since we have the mean AND the standard deviation, we can now compute the z-score.

SO! the mean is 8 and the standard deviation is .1 and X is 8.17

X?

X- Mean

Z =

Stan. Deviation

8.17 - 8

Z =

.1

The z-score is is 1.7

**Example #2**

The temperature is recorded at 60 airports in a region. The average temperature is 67 degrees Fahrenheit with standard deviation of 5 degrees. What is the z-score for a temperature of 68 degrees?

X?

mean?

Stan. deviation?

Obviously, the number of airports in the region has nothing to do with computing the z-score. We just need the X value, the mean or "average", and the standard deviation.

X - Mean

Z =

Stan. Deviation

68 - 67

z = ------

5

The Z-score is .2

**GOT THAT..??**

trY A few on your own

1. Scores on a history test have average of 80 with a standard deviation of 6. What is the z-score for a student who earned a 75 of the test?

2. Books in the library are found to have average length of 350 pages with standard deviation of 100 pages. What is the z-score corresponding to a book of length 80 pages?

3. A particular leg bone for dinosaur fossils has a mean length of 5 feet with standard deviation of 3 inches. What is the z-score that corresponds to a length of 62 inches?

1. The Z- score is -.833

75- 80

z =

5

The Z- score is - 2.7

80-350

Z =

100

The Z-Score is .667

Here we need to be careful that all of the units we are using are the same. There will not be as many conversions if we do our calculations with inches. Since there are 12 inches in a foot, five feet corresponds to 60 inches.

62-60

Z =

3

the z-score tells you how many standard deviations are above or below the mean.

First, we're going to start with z-score

Next, is Normal Distribution

SO, we learned that a density curve is an idealized description of the overall pattern of a distribution.

That density curves have many different shapes

that each region of of the density curve has a different percentage

that the median of a density curve cuts it in half

and that the mean is the "balance point" of the density curve

That the area of a density curve is 1

**and now....**

2.2 Standard Normal Calculations

**Identify the main properties of the Normal Curve**

Explain the 68-95-99.7 rule (Empirical rule)

Define the standard Normal distribution

Normalcdf..InvNorm

Explain the 68-95-99.7 rule (Empirical rule)

Define the standard Normal distribution

Normalcdf..InvNorm

**Shapes**

Lets Check out some Vocabulary

**Understanding The Empirical Rule**

The Empirical Rule ( )

68-95-99.7

The U thingy is mean (Mu) and The Fancy O is Standard Deviation (Sigma)

Ready for some examples?

Example

The scores for all high school seniors taking the verbal section of the Scholastic Aptitude Test (SAT) in a particular year had a

mean

of

490

and a

standard deviation

of

100.

The distribution of SAT scores is

bell-shaped.

A. What percentage of seniors scored between 390 and 590 on this SAT test?

B. One student scored 795 on this test. How did this student do compared to the rest of the scores?

C. A rather exclusive university only admits students who were among the highest 16% of the scores on this test. What score would a student need on this test to be qualified for admittance to this university?

A. What percentage of seniors scored between 390 and 590 on this SAT test?

From the figure above, about 68% of seniors scored between 390 and 590 on this SAT test.

B. One student scored 795 on this test. How did this student do compared to the rest of the scores?

Since about 99.7% of the scores are between 190 and 790, a score of 795 is excellent. This is one of the highest scores on this test.

Since about 99.7% of the scores are between 190 and 790, a score of 795 is excellent. This is one of the highest scores on this test.

Since about 68% of the scores are between 390 and 590, this leaves 32% of the scores outside this interval. Since a bell-shaped curve is symmetric, one-half of the scores, or 16%, are on each end of the distribution. The figure above shows these percentages.

Normal distributions are a family of distributions that have the same general shape. They are symmetric with the scores more concentrated in the m than in the tails. Notice that the differ in how spread out the are. The area under each curve is the same.

Data can be skewed to the left

or data can be skewed to the right

But there are many cases where the data tends to be around a central value with no bias left or right, and it gets close to a "Normal Distribution" like this:

A Normal Distribution

The "Bell Curve" is a Normal Distribution.

And the yellow histogram shows some data that

follows it closely, but not perfectly (which is usual

The Normal Distribution functions: NormalCDF & InvNorm..

normalcdf (

Cumulative Distribution Function

)

This function returns the cumulative probability from zero up to some input value of the random variable x.

Syntax

: normalcdf

(lower bound, upper bound, mean, standard deviation

)

invNorm (

Inverse Normal Probability Distribution Function

)

This function returns the x-value given the probability region to the left of the x-value.

Syntax:

invNorm (

probability, mean, standard deviation

)

Example

Given a normal distribution of values for which the

mean is 70

and the

standard deviation is 4.5.

Find:

a) the probability that a value is between 65 and 80, inclusive.

b) the probability that a value is greater than or equal to 75.

c) the probability that a value is less than 62.

d) the 90th percentile for this distribution.

A. Find the probability that a value is between 65 and 80, inclusive. (

This is accomplished by finding the probability of the cumulative interval from 65 to 80.

)

Syntax:

normalcdf(

lower bound, upper bound, mean, standard deviation

)

Answer: The probability is

85.361%.

Find the probability that a value is greater than or equal to 75.

(

The upper boundary in this problem will be positive infinity. Type

1 EE 99

. Enter the

EE

by pressing

2nd, comma

-- only one E will show on the screen

.)

Answer: The probability is

13.326%.

Find the probability that a value is less than 62.

(

The lower boundary in this problem will be negative infinity. Type

-1 EE 99

. Enter the

EE

by pressing

2nd, comma

-- only one E will show on the screen.

)

Answer: The probability is

3.772%.

Find the 90th percentile for this distribution.

(

Given a probability region to the left of a value (i.e., a percentile), determine the value using invNorm.

)

Answer: The x-value is

75.767.

Time to wrap it up!

Thank you for listening :)