Normal Probabilities for Personal Income 
Statistical Topic:
Once we understand the algebra (use of the function) and the calculus (area under the curve) for the Normal and Standard Normal Distributions, we can use these areas under the Standard Normal curve to compute probabilities.  We can compute these probabilities much easier using a table in your statistics textbook.
Student Issue: 
What are the probabilites that a state income will fall in a certain ranges assuming that state personal income is a variable that is normally distributed. 
Goals of Data Analysis Lab: 
Learn how to use the probability table for the Standard Normal (another name is the z-distribution).  Compute probabilites for interval of values of a theoretical Normal Distribution for state personal income.
Statistical Techniques: 
  1. For any standard normal distribution, compute the following probabilities.  Draw diagrams and shade area (probability):

    2. a) P (Z > 1.79) =
    b) P ( -1.3 < Z < 2.1) = 
    c) P(Z > -.2) = 
    d) Given the probability, find the Z-score that satisfies P(Z < z) = .18.
     
  2. Suppose the theoretical normal distribution for personal income had a mean (mu)  = $24,366 and a standard deviation (sigma) = $3755.  Find the following probabilities (sometimes called p-values).  Draw the distributions for each probability, label the parameters and shade in the probability you are computing. 

    3. a)  P (X > $25,000) =
    b)  P ( $20,000 < X < $26,000 ) = 
    c)  P( $25,000 <X < $27,000 ) = 
    d) P(X > $30,000) =
    e)  Given the probability, find the value X that satisfies P(X < x?) = .10 
     
  3. It is important to try to keep all the distributions we have discussed straight.  Take a moment and define the following pairs of terms for the state personal income variable:  Sample and Population Histogram; Normal and Standard Normal Distribution; Continuous random variables, x and z; Parameters, mu and sigma, for the Normal and Standard normal distribution.

    4.  
Social Commentary: 
  1. Why isn't the data set for personal income (see Figure 1) exactly normally distributed?