Yolande Tra: /* 95% of the Area */

2014-09-07T01:09:20Z

95% of the Area

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{{Cat|Normal Distribution| 03}}
* Areas under portions of a normal distribution can be computed by using calculus.
* Since this is a non-mathematical treatment of statistics, we will rely on computer programs and tables to determine these areas.

=Example 1=
[[File:ClipCapIt-140602-150836.PNG]]

It shows a normal distribution with
* a mean of 50
* a standard deviation of 10

The shaded area between 40 and 60 contains 68% of the distribution.

=Example 2=

:[[File:ClipCapIt-140906-210527.PNG]]

It shows a normal distribution with
* a mean of 100
* a standard deviation of 20.
As in Example 1, 68% of the distribution is within one standard deviation of the mean.

The normal distributions shown in Example1 and 2 are specific examples of the general rule that 68% of the area of any normal distribution is within one standard deviation of the mean.

=Example 3=
:[[File:ClipCapIt-140906-210401.PNG]]

It shows a normal distribution with
* a mean of 75
* a standard deviation of 10
The shaded area contains 95% of the area and extends from 55.4 to 94.6.

=95% of the Area=

* For all normal distributions, 95% of the area is within 1.96 standard deviations of the mean.
* For quick approximations, it is sometimes useful to round off and use 2 rather than 1.96 as the number of standard deviations you need to extend from the mean so as to include 95% of the area.

=Quiz=
<quiz display=simple >

{ A distribution has a mean of 40 and a standard deviation of 5. 68% of the distribution can be found between what two numbers?

|type="()"}
-30 and 50
-0 and 45
-0 and 68
+35 and 45

{
{{Show Answer|
35 and 45

68% of the distribution is within one standard deviation of the mean. 40 + 5 equals to 45, 40 - 5 equals to 35
}}
}

{A distribution has a mean of 20 and a standard deviation of 3. Approximately 95% of the distribution can be found between what two numbers?

|type="()"}
-17 and 23
+14 and 26
-10 and 30
-0 and 23

{
{{Show Answer|
35 and 45

95% of the distribution is within 1.96 standard deviations of the mean. You can round 1.96 to 2 to approximate. 20 - 2(3) equals to 14, 20 + 2(3) equals to 26
}}
}

{A normal distribution has a mean of 5 and a standard deviation of 2. What proportion of the distribution is above 3?

[http://onlinestatbook.com/2/calculators/normal.html Use Normal Calculator here]

|type="{}"}
{ 0.8413 }

{
{{Show Answer|
0.8413

Use the "Calculate Area for a given X" calculator and enter Mean of 5, SD of 2, Above 3. You will get 0.8413.
}}
}

{A normal distribution has a mean of 120 and a variance of 100. 35% of the area is below what number?

[http://onlinestatbook.com/2/calculators/normal.html Use Normal Calculator here]
|type="{}"}
{ 116.15 }

{
{{Show Answer|
116.15

Var is 100, so SD is 10. Use the "Calculate X for a given Area" calculator and enter Mean is 120, SD is 10, Shaded area is .35. Click below, and you will get 116.15.
}}
}

{A normal distribution of test scores has a mean of 38 and a standard deviation of 6. Everyone scoring at or above the 80th percentile gets placed in an advanced class. What is the cutoff score to get into the class?

[http://onlinestatbook.com/2/calculators/normal.html Use Normal Calculator here]

|type="{}"}
{ 43 }

{
{{Show Answer|
43

Use the "Calculate X for a given Area" calculator and enter Mean of 38, SD of 6, Shaded area of .80. Click below, and you will get 43.05, meaning a score of 43.
}}
}

{A normal distribution of test scores has a mean of 38 and a standard deviation of 6. What percent of the students scored between 30 and 45?

[http://onlinestatbook.com/2/calculators/normal.html Use Normal Calculator here]

|type="{}"}
{ 78.7 }

{
{{Show Answer|
78.7

Use the "Calculate Area for a given X" calculator and enter Mean of 38, SD of 6, Between 30 and 45. You will get 0.787, meaning 78.7%.
}}
}

</quiz>

Areas of Normal Distributions - Revision history

Yolande Tra: /* 95% of the Area */