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.

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>

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.

Question: Articulate this investigation as a hypothesis test assuming that the null hypothesis ca be stated as “The candidates are currently tied.”

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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>

Question: Suppose that 500 individuals are surveyed and 225 say they support Candidate A. Determine the values $n$, $X$, and $\hat{p}$ and choose the appropriate conclusion.

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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>

Solution: $n = 500$ and $X = 225$, so $\hat{p} = 0.45.$ This experimental outcome meets the rejection criterion, so we can reject the null hypothesis that the candidates are tied. Since $0.45 < 0.5$, we conclude that Candidate B is winning, so the best choice is (b).