Lesson 7 - The Normal Distribution


 

Definitions:

Continuous Variable:  A variable that can assume all values between any two given values.

Normal distribution:   A continuous, symmetric, bell-shaped distribution of a variable.
 


 
Properties of the Normal Distribution: 

1.  It is bell-shaped.

2.  The mean, median, and mode are equal and located at the center of the distribution.

3.  The curve is unimodal (only one mode).

4.  The curve is continuous; i.e., for each value of X, there is a corresponding value of Y.

5.  The curve never touches the X-axis.

6.  The area under the curve is equal to 1.00, or 100%

7.  The area under the curve that encompasses one standard deviation from the mean is 68%, within two standard deviations is 95%, and within three: 99.7%

 


 
Mean and Standard Deviation of the Normal Distribution:
 
Mean:     m = 0

Standard Deviation: s = 1

 


 
Transforming variables into the standard normally distributed variable

Simply use the Z-score formula we saw earlier in the book!

z = (x - m)/s

The z-value is actually the number of standard deviations that a particular value is away from the mean!



What does the Z-value tell us?

It yields the area under the normal distribution curve that corresponds to that value.  As will be shown, this is also used in probability calculations.

(Table 3 at the end of the book lists the area under the curve for any z-value from 0 - 3.09)

As I stated above, the area under a normal distribution curve also corresponds to a probability.  Simply transforming the four-digit decimal into a percentage yields the percent chance of the variable you’re looking at actually occurring.

 


 
A special notation is used in problems asking for probabilities using the normal distribution, and looks like this:

P(0 < z < 1.8) 
P(z < 1.29)
P(z > 2.63)
P(-1.02 > z > .82)
 

Solving problems using the Normal Distribution:

As stated before, to solve problems using the normal distribution, you first have to transform the variable to a standard distribution variable using the z-score formula.

Then-  you simply use the z-values and Table 3 (end of the text) to solve the problems given.  Be sure you change any percentage values given in the problem into decimals before beginning.

You can also work backward; when given the area under the curve, use Table 3 to find the z- variable being asked for.  Then to find the value of X,  use the formula given at the bottom.
 

X=(z times s) + m

 

 
The Central Limit Theorem

So far we have discussed single values using the Normal Distribution...but what if we want to find out properties of samples taken from a population?  We can if we make some small modifications in our formulas that we’ve seen so far.  But first, let’s look at some definitions:



Sampling distribution of sample means:  a distribution obtained by using the means computed from random samples of the same size taken from a population.

Sampling Error:  the difference between the sample mean and the corresponding population mean because the sample is not a perfect representation of the population.



But, when all the possible samples of a particular size are taken from a population, then the distribution of those sample means has three special properties:

1.  The mean of the sample means is the same as the mean of the population.

2.  The standard deviation of the sample means will be smaller than that of the population, and is called the standard error of the mean.  The formula is given at the top of page 293.

3.  As the sample size increases, the shape of the distribution of sample means will approach a normal distribution.  This is known as the Central Limit Theorem.


The formula for transforming variables from sample means into z-values is given in the middle of page 293.

So when do you use this formula over the other?

  • If you’re looking to solve a question about a sample mean, use the formula given on page 293.
  • If you’re looking to solve a question about a single data value, use the original formula we learned back in Chapter 5.
The Finite Population Correction Factor

The formula for the standard error of the mean is accurate only when samples are drawn from a very large or infinite population, or when they’re drawn with replacement to the population.  Since this seldom happens in real life we have to apply a correction factor.  This allows us to get a more accurate standard error of the mean.  The formula are listed in the green box on page 298.

*Generally, the correction factor is unnecessary if the sample size is less than 5% of the population size.
 


 
The Normal Approximation to the Binomial Distribution

When the sample size is large, it becomes very difficult to do problems using the method shown last chapter for the Binomial Distribution.  The Normal Distribution can be used to approximate the Binomial as long as the sample size (n) is large enough and the probability for success (p) is not too small.  Statisticians agree that if both n p and n q are each greater than 5, then the Normal Distribution can be used to approximate the Binomial.

The procedures for using the Normal Distribution to approximate the Binomial are shown in the text.  But you must use the correction for continuity rules given in this section to make the approximation work.  These are explained in section 5.7 of the text.

 



 
Homework:

Read sections 5.5 through 6.2.