Download Normal Distribution X ~ N(µ, σ )

 Normal Distribution
By far the most commonly used distribution in statistics. This distribution provides a good model for many continuous populations. X is a Normal Random Variable
Probability Density Function:
X ~ N(µ, σ2)
Characteristics of the Normal Distribution:
X ~ N(µ, σ2)
Symmetric
Mean = Median
65­95­99.7% Rule
Standard Units
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Each normal population can have a different mean and variance.
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We would need a different table for each to find probabilities.
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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, when dealing with normal populations, we often convert from the units in which the population items were originally measured 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)
Standard Normal Distribution
N(0, 1)
Probabilities for experiments 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)
Finding Area Under the Normal Curve
Example: Find the area under normal curve to the left of z = 0.47. 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?
Now...in your book, you will see them convert to z­scores, which is how we used to have to do it before the graphing calculators did these calculations for us. The following calculations convert the x­values to z­scores and finds the probability using the standard normal distribution. The answers are the SAME.
Problem!
Usually we will NOT know the true mean or true standard deviation of the Normal population we are working with. Woe is us...oh, what shall we do? Statistics to the rescue (again)!
We take a sample of size n. Our BEST guess at the population mean μ is _____. Our BEST guess at the population standard deviation σ is _____.
If we take all possible samples of size n from our normal population and look at the collection of sample means, then they will cluster around a center value and they will have a variance and standard deviation:
This says the sample mean is the best estimator of the true and usually unknown population mean.
There is uncertainty in using the sample mean to estimate the true unknown population mean. The uncertainty in our estimate is found by...