Ahnboyoung: /* Quiz */

2014-06-02T16:07:05Z

Quiz

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{{Cat|Normal Distribution| 04}}
Normal distributions do not necessarily have the same means and standard deviations.

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

=Area below Z=
:[[File:ClipCapIt-140602-164653.PNG]]

* The first column titled "Z" contains values of the standard normal distribution; the second column contains the area below Z.
* Since the distribution has a mean of 0 and a standard deviation of 1, the Z column is equal to the number of standard deviations below (or above) the mean.

;Example
* a Z of -2.5 represents a value 2.5 standard deviations below the mean.
* The area below Z is 0.0062.

=Formula=
A value from any normal distribution can be transformed into its corresponding value on a standard normal distribution using the following formula:
Z = (X - μ)/σ

Where:
Z is the value on the standard normal distribution,
X is the value on the original distribution,
μ is the mean of the original distribution and
σ is the standard deviation of the original distribution.

;Example
As a simple application, what portion of a normal distribution with a mean of 50 and a standard deviation of 10 is below 26? Applying the formula, we obtain
Z = (26 - 50)/10 = -2.4
* From the table above, we can see that 0.0082 of the distribution is below -2.4.
* There is no need to transform to Z if you use the applet as shown below.
[[File:ClipCapIt-140602-165355.PNG]]

=Standardizing the Distribution=
If all the values in a distribution are transformed to Z scores, then the distribution will have a mean of 0 and a standard distribution.
This process of transforming a distribution to one with a mean of 0 and a standard deviation of 1 is called standardizing the distribution

=Quiz=
<quiz display=simple>

{ A standard normal distribution has:

|type="()"}
-a mean of 1 and a standard deviation of 1
+a mean of 0 and a standard deviation of 1
-a mean larger than its standard deviation
-all scores within one standard deviation of the mean

{
{{Show Answer|
a mean of 0 and a standard deviation of 1

The standard normal distribution is defined as a normal distribution with a mean of 0 and a standard deviation of 1.
}}
}

{ A number 1.5 standard deviations below the mean has a z score of:

|type="()"}
-1.5
+-1.5
-3
-more information is needed

{
{{Show Answer|
-1.5

Z is equal to the number of standard deviations below or above the mean. Numbers below the mean have negative Z scores.
}}
}

{ A distribution has a mean of 16 and a standard deviation of 6. What is the Z score that corresponds with 25?

|type="{}"}
{ 1.5 }

{
{{Show Answer|
1.5

25 is 1.5 SDs above the mean: Z is (X - M)/SD. (25 - 16)/6 is 1.5
}}
}

{ A distribution has a mean of 18 and a standard deviation of 5. Use the table presented in this section to determine the proportion of the scores (area) below 6.

|type="{}"}
{ 0.0082 }

{
{{Show Answer|
0.0082

Z is= (X - M)/SD. (6 - 18)/5 equals to -2.40, Look at the table to see that the area below -2.40 is .0082.
}}

</quiz>
}

Standard Normal Distribution - Revision history

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