Download Normal Distribution

 Normal Distribution
By far the most commonly used distribution in statistics. Provides a good model for many continuous populations. Continuous Random Variable
Mean can be any value
Variance can be any positive value
X ~ N(µ, σ2)
X is a Normal Random Variable
Probability Distribution Function:
How will we know we have a normally distributed population?
History
Sample
Samples from normal distributions rarely contain outliers.
A boxplot can be constructed to check for outliers.
Characteristics of the Normal Distribution:
Symmetric
Mean = Median
68­95­99.7% Rule
X ~ N(µ, σ2)
Standard Units
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Each normal population can have a different mean and variance.
•
The proportion of a normal population that is within a given number of standard deviations of the mean is the SAME for any normal population. •
For this reason, we often convert to standard units. •
Standard units tell how many standard deviations an observation is from the population mean. How to Convert to Standard Units
X ~ N(µ, σ2) Z ~ N(0, 1)
Use the z­score to determine how many standard deviations away from the mean a value is.
Heights for women are normally distributed with a mean of 63.6 inches and a standard deviation of 2.5 inches.
A typical height that is more than one standard deviation above the mean.
Standard Normal Distribution
N(0, 1)
Probabilities involving the Normal Distribution can be found as areas under the Standard Normal Curve by (1) Converting x­values to z­scores and using the Standard Normal table in your book. (2)
Using the "normalcdf(" function on your graphing calculator
normalcdf(lowx, highx, mean, standard deviation)
TI­84 use ­99999 if there is no lower limit. TI­89 has an infinity button.
Finding Area Under the Normal Curve
Example: Find the area under normal curve to the left of z = 0.47. Example: Find the area under normal curve to the right of z = 0. Finding Area Under the Normal Curve
Example: Find the area under the curve to the right of z = 1.38.
Finding Area Under the Normal Curve
Example: Find the area under the curve between z = ­2.3 and z = 1.5.
Example: A process manufactures bolts whose diameters are normally distributed with mean 1.5 cm and standard deviation 0.1 cm. What percentage of bolts manufactured fall between 1.25 and 1.65 cm?
A bolt is selected at random. What is the probability it has a diameter greater than 1.6 cm?
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Five bolts are selected at random. What is the probability three of them have a diameter greater than 1.6 cm?
What is the 25th percentile of the standard normal distribution?
What diameter corresponds to the 75th percentile of a N(10, 2)?
Combining Normal Random Variables