Null hypothesis¶
Problem
According to the survey, the candidate of party A would have received about 50% of the votes if the election had taken place at the time of the survey. A success at the first ballot, for which more than 50% of all votes are required, is hence questionable. Thus, the election campaign consultant put in place by party A suggests an additional campaign in the final stage of the election battle. However, the treasurer of party A would prefer to avoid the high costs caused by an additional campaign, if possible.
In order to come to a decision on the realization of an additional campaign, the null hypothesis “The candidate of party A would currently receive at most 50% of all votes.” is to be tested by means of a sample of 200 eligible voters on a level of significance of 5%. Determine the associated decision rule.
Justify that the choice of the null hypothesis for the described test is in accordance with the concern of the election campaign consultant to achieve a success already at the first ballot.
Solution of part a
We want to disprove the null hypothesis. For that we assume that 50% of the voters vote for the candidate of party A. In a survey of 200 people, we have to determine the number \(k\) of people who vote for our candidate such that the level of significance is 5%. Thus, the equation
has to be solved for \(k\). From a mathematical table for the binomial distribution, we can determine \(k\approx112\). Alternatively, we can use Sage:
Furthermore, we can simulate the survey and check in how many surveys at least 112 people would indicate to vote for candidate A, although the probability that a person votes for candidate A is 50%.
Solution of part b
With the chosen null hypothesis one can relatively safely say that with at least 112 positive statements the candidate of party A will be elected. If the first survey is correct about the candidate receiving only about 50% of the votes, the null hypothesis will probably be disproven and the funds for an additional campaign get approved.
