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.

- Present the data in an appropriate form. Report the sample size.
- Obtain a 95% confidence interval for the proportion of all likely voters prefering each of the three leading candiates.
- Along with the data summary called for in part (a), report the blanket error margin.
- 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? - 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%.

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).

Candidate | % Favoring |
---|---|

Algore | 31.9 |

Bushert | 26.1 |

Cain | 22.6 |

Others | 6.6 |

Undecided | 12.8 |

n = 986, Error Margin ± 2.9% |

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

Candidate | 95% CI |
---|---|

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 H

_{0}: P_{C}= .25 against H_{A}: P_{C}< .25 where P_{C}is theresult for Dr. Cain (this is the proportion of all voters who will vote for McCain). The test statistic computes to -1.74,__election__*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 H

_{0}: 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 H_{0}at the 5% significance level. This race*is not*a statistical tie.