Suppose there is an election involving two individuals: Candidate A and Candidate B. Candidate A hires a polling firm to assess how things are going. The candidate’s team decides that if either candidate has more than 52% support in the survey, then they will assess that the candidate has a substantial lead in the race.

Suppose that 500 individuals are surveyed and 255 say they support Candidate A. Which of the following is the appropriate conclusion for the candidate’s team?

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Multiple Choice

1. This meets the team’s criterion to reject the null hypothesis. They conclude that Candidate A is winning the race.
2. This meets the team’s criterion to reject the null hypothesis. They conclude that Candidate B is winning the race.
3. This does not meet the team’s criterion to reject the null hypothesis. They conclude that neither candidate has a substantial lead.
4. This does not meet the team’s criterion to reject the null hypothesis. They conclude that they do not have sufficient evidence to draw a conclusion. </aside>

Method of solution.

In the end, we will show that (d) is the correct answer.

Prompt 1: What is the question?

In plain language, establish an appropriate null hypothesis for this situation. What should be the associated alternative hypothesis?

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• Null Hypothesis: The candidates are tied among eventual voters.
• Alternative Hypothesis: The candidates are not tied among eventual voters. </aside>

Prompt 2: Establish mathematical notation

Parameter of interest: $p$, the true proportion of eventual voters who support Candidate A.

Let $\hat{p}$ represent the proportion of individuals in the survey who say they support Candidate A.

Let $n$ represent the number of individuals who participate in the survey.

Let $X$ represent the number of individuals in the survey who say they support Candidate A.

Prompt 3: Write the question in a hypothesis test structure. The natural null hypothesis is: “The candidates are currently tied,” which corresponds to $p = 0.5$.

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Hypothesis Test

• Null Hypothesis: $H_0: p = 0.5$
• Alternative Hypothesis: $H_A: p \neq 0.5$
• Test Statistic: $TS: \hat{p} = X/n$
• Rejection Criterion: $\hat p < 0.48$ or $\hat p > 0.52$. </aside>

Prompt 4: Determine the values $n$, $X$, and $\hat{p}$ and choose the appropriate conclusion.

We have that $n = 500$ and $X = 255$, so $\hat{p} = 0.51.$ This experimental outcome does not meet the stated rejection criterion. So the team cannot reject the null hypothesis. HOWEVER, also note that we can never prove the null hypothesis is true. (A survey with a larger sample size might be more powerful and show that the null hypothesis is indeed false.) So we cannot say affirmatively that the candidates are tied, only that we cannot rule out that they are tied. The best answer is (d).