Exercises About Election Polls

Two background documents are available:

• What constitutes a statistical tie (no technical details, merely discussion and a simple illustration of the method).

• How to ascertain a statistical tie (including some technical details).

In addition, there are questions about forming confidence intervals and computing P-values for a single proportion. This topic is covered in your textbook.

Questions

The village of Ogewso looks forward to electing a mayor. A random sample is drawn from the group of likely registered voters. Among the polled individuals, 315 prefer Mr. Algore, 257 prefer Mrs. Bushert, 223 prefer Dr. Cain, 65 prefer one of a number of other candidates, leaving 126 undecided.

1. Present the data in an appropriate form. Report the sample size.
2. Obtain a 95% confidence interval for the proportion of all likely voters prefering each of the three leading candiates.
3. Along with the data summary called for in part (a), report the blanket error margin.
4. Dr. Cain has indicated that she will pull out of the race early if it becomes clear that her election result will be below 25% of the voting public. Maybe she ought to pull out now? Test the appropriate hypotheses for making this determination. State the hypotheses, compute the test statistic and obtain the P-value. Based on the statistical analysis, does it appear as though she should quit now? What other factors might play a role in her decision?
5. Are Algore and Bushert in a "statistical tie?" Test the appropriate hypotheses for making this determination. State the hypotheses, compute the test statistic and obtain the P-value. Make your decision with the significance level set at 5%.

Solutions

There are n = 986 individuals polled.

1. The proper way to summarize these results is with a simple table of percentages. (Don't use pie charts, nor bar graphs--they stink.) I've included the answer for part (c) (the blanket error margin).

2. Here you see 95% CIs for each of the three candidates.

3. Use the largest error margin for the blanket. This always corresponds to the category including closest to 50% of the observations -- in this case the Algore category.

4. To answer we need to test H₀: P_C = .25 against H_A: P_C < .25 where P_C is the election result for Dr. Cain (this is the proportion of all voters who will vote for McCain). The test statistic computes to -1.74, P-value is .0409. This is small (less than the standard 5% level ofter used to make decisions). It indicates that less than 25% of likely voters prefer Dr. Cain:.The statistics indicate there's strong evidence Dr. Cain's goal won't be attained. However, note the large percentage of "Undecided" voters. If Cain is able to attract a large fraction of these voters, and maybe even change the minds of some voters who currently do have another preference, he may well not only get that 25% -- he may even win this election.

5. Here we need to test H₀: P_A - P_B = 0 against H_A: P_A - P_B ¹ 0, where P_A and P_B are the election results for Mr. Algore and Mrs. Bushert respectively. The estimated difference is .0588. The standard error computes to be .0243. The test statistic is Z = .0588/.0243 = 2.42. Tail area: .0078. P-value = .0156 or 1.56%. Decision: Reject H₀ at the 5% significance level. This race is not a statistical tie.