Question

Influencing Voters When getting voters to support a candidate in an election, is there a difference between a recorded phone call from the candidate or a flyer about the candidate sent through the mail? A sample of 500 voters is randomly divided into two groups of 250 each, with one group getting the phone call and one group getting the flyer. The voters are then contacted to see if they plan to vote for the candidate in question. We wish to see if there is evidence that the proportions of support are different between the two methods of campaigning. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Possible sample results are shown in Table 4.3 . Compute the two sample proportions: \(\hat{p}_{c},\) the proportion of voters getting the phone call who say they will vote for the candidate, and \(\hat{p}_{f},\) the proportion of voters getting the flyer who say they will vote for the candidate. Is there a difference in the sample proportions? (c) A different set of possible sample results are shown in Table 4.4. Compute the same two sample proportions for this table. (d) Which of the two samples seems to offer stronger evidence of a difference in effectiveness between the two campaign methods? Explain your reasoning. $$ \begin{array}{lcc} \hline & \begin{array}{c} \text { Will Vote } \\ \text { Sample A } \end{array} & \text { for Candidate } & \begin{array}{l} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 152 & 98 \\ \text { Flyer } & 145 & 105 \\ \hline \end{array} $$ $$ \begin{array}{lcc} \text { Sample B } & \begin{array}{c} \text { Will Vote } \\ \text { for Candidate } \end{array} & \begin{array}{c} \text { Will Not Vote } \\ \text { for Candidate } \end{array} \\ \hline \text { Phone call } & 188 & 62 \\ \text { Flyer } & 120 & 130 \\ \hline \end{array} $$

Short answer

Based on the analysis, Sample B provides stronger evidence that the two campaign methods do not have the same effectiveness. The difference between the proportions of voters that will vote for the candidate under the phone call method and under the flyer method is larger in Sample B than in Sample A.

Step-by-step solution

Define Parameters

To analyze the given scenario, let's define the relevant parameters. We have \(p_c\) as the proportion of voters who received a phone call and said they would vote for the candidate, and \(p_f\) as the proportion of voters who received a flyer and said they would vote for the candidate.

State Hypotheses

We need to state our null hypothesis (\(H_0\)) and alternative hypothesis (\(H_1\)). \(H_0\) would be \(p_f = p_c\), meaning there is no difference in the proportions of those who would vote for the candidate between the two campaign methods. \(H_1\) would be \(p_f ≠ p_c\), signifying there is a difference in the proportions of those who would vote for the candidate among the two campaign methods.

Compute Sample Proportions: Sample A

We compute the proportion of voters who plan to vote for the candidate in both the phone calls and flyers methods, with data from Sample A. For phone calls, \(\hat{p}_{c}A = 152/250 = 0.608\) and for flyers, \(\hat{p}_{f}A = 145/250 = 0.58\). Thus, the proportion of voters that will vote for the candidate seems slightly higher for phone calls in Sample A.

Compute Sample Proportions: Sample B

We do a similar calculation for Sample B. For phone calls \(\hat{p}_{c}B = 188/250 = 0.752\) and for flyers, \(\hat{p}_{f}B = 120/250 = 0.48\). The proportion of voters that will vote for the candidate is significantly higher for phone calls in Sample B.

Compare Results

Comparing the two samples A and B, it appears that Sample B provides stronger evidence for a difference in effectiveness between the two campaign methods. The difference between the proportions \(\hat{p}_{c}\) and \(\hat{p}_{f}\) in Sample B is significantly larger than in Sample A.

Key concepts

Hypothesis Testing in Campaign Methods

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. In the context of campaigning methods, hypothesis testing is used to determine if one method, like a phone call, is more effective than another method, such as sending a flyer, in influencing voters' decisions.

Let's apply this to an election campaign scenario. We collect data from a sample of voters who have experienced each of these two campaigning techniques. Then, we perform hypothesis testing to decide if there is a statistically significant difference between the success rate of these methods or not. This process involves defining both a null hypothesis and an alternative hypothesis, carrying out calculations to find sample proportions, comparing these proportions, and then making a decision based on statistical evidence.

Understanding Sample Proportions

Sample proportions are statistics that provide estimates of the true proportion within the entire population. For example, when you're investigating the effectiveness of campaign methods, you calculate the sample proportion by dividing the number of voters who respond positively to a method by the total number of voters sampled for that method.

In our case, the sample proportions, denoted as \(\hat{p}_{c}\) for phone calls and \(\hat{p}_{f}\) for flyers, are essential for comparing the effectiveness of the two campaign methods. These proportions help us to quantify the level of support for the candidate and determine if there's a notable difference between the impacts of phone calls and flyers. Understandably, computing these correctly is paramount for drawing reliable conclusions from our hypothesis test.

Null and Alternative Hypotheses

In any hypothesis testing situation, we start by stating the null hypothesis, often denoted as \(H_0\), and the alternative hypothesis, denoted as \(H_1\) or \(H_a\). The null hypothesis is a statement of no effect or no difference. It's the hypothesis that we assume to be true unless the sample data provide strong evidence against it.

In the election campaign example, the null hypothesis states that there is no difference in voter support between those contacted by phone call and those who received a flyer, \(p_f = p_c\). Conversely, the alternative hypothesis posits that there is a difference, \(p_f eq p_c\). During the analysis, we check whether the sample data provide enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Assessing Statistical Evidence

Assessing statistical evidence involves analyzing sample data to decide whether or not they support the null hypothesis. If the evidence strongly contradicts our null hypothesis, we might reject it in favor of the alternative hypothesis.

In our scenario with the campaign methods, we compare the sample proportions from two different sets of sample results. If the difference between the sample proportions \(\hat{p}_{c}\) and \(\hat{p}_{f}\) is substantial, it implies that one method may truly be more effective than the other. Statisticians typically use a significance level to determine if the observed evidence is strong enough to reject the null hypothesis. The concept of p-value also plays a crucial role here – if the p-value is less than our significance level, we reject the null hypothesis indicating that our statistical evidence points to a difference in campaign methods' effectiveness.