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.
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.
There are n = 986 individuals polled.
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).
|n = 986, Error Margin ± 2.9%|
Here you see 95% CIs for each of the three candidates.
|Algore||31.9% ± 2.9%|
|Bushert||26.1% ± 2.7%|
|Cain||22.6% ± 2.6%|
|Others||6.6% ± 1.5%|
|Undecided||12.8% ± 2.1%|
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.
To answer we need to test H0: PC = .25 against HA: PC < .25 where PC 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.
Here we need to test H0: PA - PB = 0 against HA: PA - PB ¹ 0, where PA and PB 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.