"As the sample size increases, the sampling distributions of sample means approaches a normal distribution."
 
 
 
 
 
 
 

    In class, have pairs of students generate set of random numbers from 1-10, with one generating, the other entering in a list.  Find the mean and standard deviation. Bring the results of the individual ones to the overhead calculator.   The class totals for numbers 1..10 will be plotted and  and  calculated.  The Averages will also be plotted and  and  calculated
 
 
 
 
 
 
 
 
 
 
 
 

 Show overhead transparencies, that as , the sample size increases, the histogram of becomes "mound shaped".  Pictures pg. 256.
 
 
 
 
 
 
 
 
 

 No matter WHAT the original population, if  n>30 the distribution of  is approximated normal.
 

    If the original population is normal, the means are ALWAYS normal.
 
 
 

CENTRAL LIMIT THEOREM  page 257
 
 
 

NOTATION 

        Mean of the Sample Means    
 
                                        

        Standard Deviation of Sample Means

     often called STANDARD ERROR OF MEAN

EXAMPLES
#4    Mean = 63.6  Standard Deviation = 2.5

        a)  A single individual                 

            P(63.0 < x < 65.0)
 
 
 
 
 
 

       b)  SAMPLE Mean       n = 75
 
                                                  

                   P(63.0 <       <65.0)
 
 
 
 
 
 
 
 
 
 
 
 
 

#10   Mean = 32.473    Standard Deviation = 5.601

        a)  P(x < 29.000)
 
 
 
 
 
 
 
 
 

      b)  P(  < 29.000)
 
 
 
 
 
 
 
 

      c)  Original population is normal
 
 
 
 
 

** OPTIONAL  EXTRA CREDIT ONLY
Finite POPULATION CORRECTION FACTOR:
Only in problems 21,22 & 24  -
    IF sample without replacement and  is greater than 5% of the the finite population size , adjust the Standard Error of Mean by multiplying as above.