Normal+Distribution

DEFINITION In many natural processes, random variation conforms to a particular probability distribution known as **NORMAL DISTRIBUTION,** which is the //most commonly used probability distribution.// The normal distribution is pattern for the distribution of a set of data which follows a bell shaped curve. Normal distribution is frequently represented in the form of a histogram. This distribution is sometimes called the __Gaussian distribution__ in honor of Carl Friedrich Gauss, a famous mathematician.

The NORMAL DISTRIBUTION EQUATION is: //Y is a normal random variable// //X is a normal random variable// //μ is the mean// //σ is the standard deviation//

**Understanding Normal Distribution**
-Normal distribution is the pattern for the distribution of a set of data and involves probability and the chances that a variable will have a certain value -**//X axis represents the value of the variable//** -//**Y axis represents the probability that the value will occur**// -When looking at the bell shaped curve of normal distribution, the area underneath the curve is associated with the probability
 * A large area underneath the curve implies a large probability.
 * A small area underneath the curve implies a small probability.

Normal distribution implies the following things regarding probability...
 * The total area under the normal curve = 1 because there is a 100% probability that the value is from the set of all real numbers
 * The probability that a normal random variable //X// equals any particular value is 0
 * The probability that //X// is greater than //a// equals the area under the normal curve bounded by //a//and plus infinity (as indicated by the //non-shaded// area in the figure below)
 * The probability that //X// is less than //a// equals the area under the normal curve bounded by //a// and minus infinity (as indicated by the //shaded// area in the figure below)



The normal distribution with a mean of 0 and and a standard deviation of 1 is very important. We refer to this distribution as the **standard normal distribution**.

THE NORMAL CURVE All normal distributions look like a symmetric, bell-shaped curve The graph of normal distribution is dependent upon two things- the mean (μ) and the standard deviation (σ) **Mean** determines the location of the center of the graph - when standard deviation is __large__, the curve is short and wide (figure b) - when the standard deviation is __small__, the curve is tall and narrow (figure a) a.b.
 * Standard deviation:** a measure of how spread out your data are

Also, there are a few rules regarding the probability of the normal curve, known as the __empirical rule__ or the __68-95-99.7 rule__. They are as a follows: - About 68% of the area under the curve falls within one standard deviation of the mean - About 95% of the area under the curve falls within two standard deviations ofthe men - About 99.7% of the area under the curve falls within three standard deviations of the mean

**Example 1** The amount of mustard dispensed from the machine at the //Hotdog Emporium// is normally distributed with a mean of 0.9 ounce and a standard deviation of 0.1 ounce. If the machine is used 500 times, approximately how many times will it be expected to dispense 1 or more ounces of mustard? a) 5 b) 16 c) 80 <span style="color: #000000; font-family: 'Trebuchet MS',Verdana,Arial,Helvetica,sans-serif; font-size: 10pt;">d) 100

//Solution:// **The correct answer is C.** <span style="color: #ff0000; font-family: Arial,Helvetica,sans-serif; font-size: 90%;">The mean is 0.9 and the standard deviation is 0.1. If one standard deviation is added to the mean, the result is 1.0 ounce. Therefore, dispensing 1 or more ounces falls into the category above one standard deviation to the right of the mean. Reading from the bell curve chart, 15.9% of data falls at or above 1 standard deviation. 15.9% x 500 = 79.5 or //approximately// 80 times to dispense one or more ounces of mustard.

<span style="color: #000000; font-family: 'Trebuchet MS',Verdana,Arial,Helvetica,sans-serif; font-size: 10pt;">**Example 2** <span style="color: #000000; font-family: 'Trebuchet MS',Verdana,Arial,Helvetica,sans-serif; font-size: 10pt;">An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most 365 days?

<span style="color: #ff0000; font-family: 'Trebuchet MS',Verdana,Arial,Helvetica,sans-serif; font-size: 10pt;">//Solution:// Given a mean score of 300 days and a standard deviation of 50 days, we want to find the cumulative probability that bulb life is less than or equal to 365 days. Thus, we know the following: <span style="color: #ff0000; font-family: 'Trebuchet MS',Verdana,Arial,Helvetica,sans-serif; font-size: 10pt;">We enter these values into the Normal Distribution Calculator and compute the cumulative probability. The answer is: P( X __<__ 365) = 0.90. Hence, there is a 90% chance that a light bulb will burn out within 365 days.
 * <span style="color: #ff0000; font-family: 'Trebuchet MS',Verdana,Arial,Helvetica,sans-serif; font-size: 10pt;">The value of the normal random variable is 365 days.
 * <span style="color: #ff0000; font-family: 'Trebuchet MS',Verdana,Arial,Helvetica,sans-serif; font-size: 10pt;">The mean is equal to 300 days.
 * <span style="color: #ff0000; font-family: 'Trebuchet MS',Verdana,Arial,Helvetica,sans-serif; font-size: 10pt;">The standard deviation is equal to 50 days.

A machine is used to fill soda bottles. The amount of soda dispensed into each bottle varies slightly. Suppose the amount of soda dispensed into the bottles is normally distributed. If at least 99% of the bottles must have between 585 and 595 milliliters of soda, find the greatest standard deviation, to the nearest hundredth, that can be allowed.
 * Example 3**

//Solution:// The 99% implies a distribution within 3 standard deviations of the mean. the difference from 585 milliliters to 595 milliliters is 10 milliliters. symmetrically divided, there are 5 milliliters used to create 3 standard deviations on one side of the mean. Dividing 5 by 3, we get the standard deviation to be 1.67 milliliters, when rounded to the nearest hundredth.

Work Cited http://www.childrensmercy.org/stats/definitions/norm_dist.htm http://stattrek.com/probability-distributions/normal.aspx http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalPrac.htm