Math 3318, Section 4
Spring 2002
Class exercises for Wednesday, May 8

Introduction. This is a revised version of the supplementary class exercises that we didn't have time for on Monday. It includes a simplified method for solving the problems. We'll also see that the simplification doesn't affect the accuracy very seriously for this type of problem.

  1. In a city in the Upper Midwest, candidates  R  and  S  are the two finalists in the mayoral election. A local TV station wants to be the first to predict the outcome. Accordingly, reporters are sent to do exit polling of voters at a representative sampling of precincts. They interview 100 voters, of whom 57 say that they voted for  R  and 43 say that they voted for  S.
     
         While the station wants to be first, they also don't want to be wrong. They decide to announce their prediction only if they have a 90% chance of being correct, i.e. only if the results indicate a 90% or better chance that the apparent winner was indeed the choice of more than 50% of the voters. In this series of exercises, we're going to help them figure this out.
     
    1. Let  p  be the proportion of voters who voted for candidate  R.  Given that 100 voters were polled, find a formula for the expected number (or mean)   of these 100 voters who voted for  R,  as a function of  p.
       
      For the standard deviation, we have the usual formula  . In an election that's not badly lopsided, we can take  p = 1/2 as an approximation. (The accuracy of this approximation will be discussed below.)
      We then have  q = 1 - p = 1/2,  and with  n = 100,   the formula gives  s = 5.
       
    2. Use Table 11.6 in the text to find  z  such that 90% of the outcomes in a normal distribution are at or below  z.
       
    3. Given the values of  ,  s,  and  z  from the previous exercises, it is 90% certain that the value of  p  will be such that:
                            + zs > 57                      (*)
      Substitute the values of  ,  s,  and  z  from exercises a) and b) above into the inequality  (*)
      to get an inequality in which  p  is the only unknown.
       
    4. Solve the inequality to determine the range of values of  p  for which your inequality is satisfied.
       
    5. Is it at least 90% certain that candidate  R  has won the election?
       
       
  2. In another Upper Midwest city, candidates  B  and  K  are in a somewhat closer mayoral race. When a representative sample of 100 voters is polled, 53 say that they voted for  K,  and 47 say that they voted for  B.  In this exercise, you may use the same formula for    that was obtained in the previous exercise, as well as the numerical values of  s  and  z.
     
    1. Find an inequality, similar to inequality (*) in the exercise 1, which is satisfied with 90% certainty.
      Suggestion: only one thing in the inequality needs to be changed.
       
    2. Substitute the values of  ,  s,  and  z  from exercise 1 to get an inequality in which  p  is the only unknown.
       
    3. Using the same method as in exercise 1, solve the inequality to determine the range of values of  p  for which your inequality is satisfied.
       
    4. Is it at least 90% certain that candidate  K  has won the election?
       
       
  3. Realizing that they can't yet call this second election, and with a small amount of time left before the polls close, the news department sends out an additional crew of reporters. They manage to poll a total of 400 voters, of whom 213 (or 53.25%) say that they voted for candidate  K.
     
    1. Find new values of   and  s.   (Remember that  n = 400  now.)
    2. Find an inequality, similar to inequality (*) in the exercise 1, which is satisfied with 90% certainty.
       
    3. Substitute appropriate values of  ,  s,  and  z  to get an inequality in which  p  is the only unknown.
       
    4. Solve the inequality to determine the range of values of  p  for which your inequality is satisfied.
       
    5. With this additional data, it is now at least 90% certain that candidate  K  has won the election?
       
       

Concerning the accuracy of the approximation  s = 1/2

       In order to avoid having the unknown  p  inside the square root sign, we sustituted  p = 1/2  in the forumla  .  This led to the somewhat simplified formula  s = 1/2. What we claim about this is that it's a reasonably good approximation, provided that  p  isn't too far from .5.

      To see whether this is really valid, let's tabulate a few values:

  p       q = 1 - p      
  .5  .5   = .5 
  .6  .4   = .49 
  .7  .3   = .46 

The tabulation supports our claim about the approximation being reasonably good. Note that it does depend on  p  being reasonably close to .5. Thus, for instance, if the value of  p  had been .1 or .9, the quantity inside the square root would have been  .09n,  and  s   would have been  .3.  A value like that would be comparable to the first version of the walleye problem (exercise 11.5.13 in the text), where we obtained very imprecise estimnates of the number of fish in the lake.

   

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