Review Requirements for a Binomial Distribution

•     Fixed Number of Trials
•     Each Trial independent
•     Each trial 2 possible outcomes
•     Probabilities remain constant    p  and  q = 1 - p

If       and      , then the binomial random variable is approximately normally distributed with the mean and standard deviation of    and 

Using a CONTINUOUS  model to Approximate a DISCRETE
 

WHEN TO USE -

• Prefer technology if technology will compute binomials sums directly
• If large value of n, or large number to sum, normal is appropriate
• MUST CHECK IF       and      to see if appropriate to use Normal approximation

    #2
 
 
 
 
 
 

    #4
 
 
 
 
 

   #6
 
 
 
 
 

    #8
 
 
 
 
 

Procedure for Normal Approximation to Binomial
            Page 267, especially 5...9

Example

    24% of people have answering machines.  Sample 2500 households.  What is the probability that more than 650 have answering machines.

            P  =  651+652+653+...+2499+2500
 
 
 
 
 
 

        Test if Normal is appropriate

                np = .24 X 2500 = 600
                nq = .76 X 2500 = 1900
                    and both are greater than 5

 = np = 600 =                    =21.4

        P(N > 650)  =
 
        P(x  > 650.5) =

        P( z  >
 
 

#10       Use program

            Test if Normal appropriate
 
 
 

#18        p = .66,   n = 1000,    Test if Normal appropriate

 =

            P(x > 699.5) =

            P(z >
 
 
 
 

#22        p = .07,   n = 250,

    =

            P(x < 4.5) =

        P(z <
 
 

#28        p = .10,    n = 50,

  =

        P(x > 1.5)  =

    P(z >
 
 
 
 
 

DISCUS THE "LANGUAGE" of these problems versus sections 5.2, 5.3, 5.4, and 5.5.  No Mean and Standard Deviation are given  ...  probability or percentage given ... and we must compute mean and standard deviation.  DON'T FORGET the continuity correction factor.