Class Notes Ch 6 Sec 1 thru 3 - Ms. Zaleski's Math …mszaleski.weebly.com/uploads/1/3/1/5/13152548/class...Class Notes Ch 6 Sec 1 thru 3.pptx Author Terry Zaleski Created Date 5/1/2016

+

The Normal Distribution

Chapter 6

+Normal Distributions Chapter 6, Section 1

+Normal Distributions

!  In Chapter 5, we drew histograms representing probability distributions for discrete random variables.

!  Instead let’s use a continuous variables such as height or weight of an adult woman. If we construct a histogram with the data for our sample, we might find that it is normally distributed.

!  A normal distribution is also called a bell curve.

!  A normal distribution can be used to describe many different variables.

!  We will learn about the properties of the normal distribution in this chapter.

+The Shape of the Normal Distribution

+The Definition of a Normal Distribution

!  If a random variable has a probability distribution whose graph is continuous, bell-shaped, and symmetric, it is called a normal distribution.

!  The graph of a normal distribution is a normal distribution curve.

!  The specific shape and position of a normal distribution curve depends on two things: !  The mean, μ

!  The standard deviation, σ

+The Shape of a Normal Distribution

!  The center of the normal distribution is the mean, μ.

!  The spread of the normal distribution is determined by the standard deviation, σ.

+A Normal Distribution Comparison

!  Which curve has the greater mean?

!  Which curve has the greater standard deviation?

+Normal Distribution Curves

!  Consider the normal curves shown at the right.

!  Which curve has the greatest mean?

!  Which curve has the greatest standard deviation?

+Skewed Distributions are NOT Normal

+The Normal Distribution

!  The mean, median and mode are equal and are located in the center of the distribution.

!  The normal curve is bell-shaped.

!  The normal curve is symmetric about the mean, μ.

!  The normal curve is continuous – there are no gaps or holes.

!  The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from from the mean.

!  The total area under the normal curve is 1.00 or 100%.

Characteristics

+The Normal Distribution

!  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 from curving upward to curving downward are called inflection points.

!  Inflection points always occur at μ-σ and μ+σ.

Characteristics

+Examples of Normal Curves

!  Notice that the mean, μ, is always the center.

!  Notice that the points of inflection always occur at μ-σ and μ+σ.

+Recall the Empirical Rule…

"  The area under the normal curve totals 1.00 or 100% "  Recall that approximately 95% of the data values fall within μ± 2σ.

"  This means that the area under the curve between μ- 2σ and μ+ 2σ is 0.95 or 95%.

+The Standard Normal Distribution

!  The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

!  The way we convert a normal distribution to a standard normal distribution is by using a z-score.

+The Standard Normal Distribution

!  Remember that a z-score or z-value (also called standard score) is the number of standard deviations the data value is from the mean.

!  Recall from chapter 3, that we used the following formula for a z-score:

!  We will now calculate the z-values for a population using the following formula:

!  Once the data values (X values) are transformed, using this formula, they become z-values.

z-values or z-scores

z =X − Xs

z =X − µ

σ

+The Standard Normal Distribution

!  The area under the standard normal curve equals 1.00 or 100%.

!  The area in any particular “section” of the standard normal curve represents the probability that a data value falls in that section.

+The Standard Normal Distribution

!  We can use Table E (p.788) to find the area to the left of any z-value between -3.49 and 3.49.

"  The area is close to 0 for z-scores close to z = -3.49.

"  The area increases as z-scores increase.

"  The area at z = 0 is 0.5 or 50%.

"  The area is close to 1 for z-scores close to z = 3.49.

+Example: Finding Area to the LEFT

"  Step 1: Draw a standard normal curve and shade the area.

"  Step 2: Look up the z-value in the table and use the area given.

+Example: Finding Area to the RIGHT

"  Step 1: Draw a standard normal curve and shade the area.

"  Step 2: Look up the z-value in the table and subtract the area from 1.

+Example: Finding Area BETWEEN any TWO z-values

"  Step 1: Draw a standard normal curve and shade the area.

"  Step 2: Look up the two z-values in the table and subtract the small from the large.

+Notation…

!  Recall that the area under the standard normal curve for a particular interval is the probability that a data value falls in that interval.

!  For example if we want the area under the curve that falls between the z-values of 0 and 2.32, we are referring to the probability that a particular data value will fall between these two z-values.

!  We can write this probability as P(0 < z < 2.32)

+Examples: Finding the Area Under the Standard Normal Distribution

+Example: Finding a z-value that Corresponds with an Area

+Applications of the Normal Distribution Section 6-2

+The Standard Normal Distribution and Practical Applications

!  We can convert any variable that in normally distributed into a standard normal distribution variable (z-value ) by using the following formula.

!  Consider a standardized test that has a mean of 100 and a standard deviation of 15. When the scores are transformed to z-values, the two distributions coincide.

z =X − µ

σ

+Steps For Finding the Area Under Any Normal Curve

Step 1: Draw a normal curve and

shade the desired area.

Step 2: Convert the values of X

(data values) to z-values using the

formula.

Step 3: Find the corresponding

area, using Table E on page 788.

z =X − µ

σ

+Example of an Application

+Example of an Application

+Example of an Application

+Finding Data Values Given Specific Probabilities

!  Formula for finding a z-value is

!  Formula for find a data value (X) is

z =X − µ

σ

X = z ⋅ σ + µ

+Finding Data Values for Specific Probabilities

Step 3: Calculate the X value by using the formula X = zσ + μ

Step 2: Find the z-value from Table E that corresponds to the desired probability (area).

Step 1: Draw a normal curve and shade the desired area that represents the probability, proportion or percentile.

+Example of an Application

+Example of an Application

+Determining Normality

!  We may want to check if a distribution that we are working with is a normal distribution so that we can use the methods learned in this chapter.

!  One way to check for normality is to draw a histogram for the data and check its shape. !  If the distribution is not approximately bell-shaped, the data are not

normally distributed.

!  If the data is skewed, then it is NOT normally distributed. There is also a test to check if the data is skewed…

!  Look for outliers. Recall that outliers are outside the range of Q1 – 1.5(IQR) and Q3 + 1.5(IQR) where the IQR = Q3 – Q1. The existence of outliers can have a big effect on normality.

+The Pearson Coefficient

!  The Pearson coefficient (PC) can be calculated as follows:

A Test for Skewness

PC =3 X − median( )

s

PC ≥ 1 Data significantly skewed right

PC ≤ -1 Data significantly skewed left

PC = 0 Data not skewed

+Example of Determining Normality

+Example of Determining Normality