Question

Congress passed the Health Care Reform Act in 2010 that will require all small businesses to have a health plan in place by 2014 and included the provision that no one could be denied coverage due to pre-existing conditions. A Newsweek Poll based on \(n=848\) registered voters nationwide asked respondents to "Please tell me if you think the health care reform law passed earlier this year was good for the country or bad for the country in general." \({ }^{4}\) The data that follow are based on the results of this poll. a. Are there significant differences in the proportions of those surveyed who think the health care reform law is good, bad, or unsure among the Republicans, Democrats, and Independents? Use \(\alpha=.05 .\) b. If significant differences exist, describe the nature of the differences by finding the proportions of those who think the law is good, bad, or unsure for each of the given affiliations.

Short answer

Answer: To determine if there are significant differences in the opinions among the three groups, we would perform a hypothesis test using a chi-square test. If the calculated chi-square statistic is greater than the critical value at a significance level of 0.05 with 2 degrees of freedom, we would reject the null hypothesis and conclude that there are significant differences in the proportions among Republicans, Democrats, and Independents.

Step-by-step solution

State the Null and Alternative Hypotheses

The null hypothesis (\(H_0\)) states that there is no difference in the proportions among Republicans, Democrats, and Independents who think the health care reform law is good, bad, or unsure. The alternative hypothesis (\(H_a\)) states that there is a difference in the proportions among the three groups. \(H_0: p_{R} = p_{D} = p_{I}\) \(H_a: \) At least one proportion differs from the others

Calculate the Expected Proportions

To calculate the expected proportions, first find the overall proportions of those who think the law is good, bad, or unsure, and then calculate the expected proportions for each group based on these overall proportions. Let \(n_R, n_D, n_I\) denote the number of Republicans, Democrats, and Independents, and \(x_{G,R}, x_{G,D}, x_{G,I}, x_{B,R}, x_{B,D}, x_{B,I}, x_{U,R}, x_{U,D}, x_{U,I}\) the number of good, bad and unsure opinions in each group. Calculate the overall proportions, \(P_G, P_B, P_U\), for good, bad, and unsure opinions: \(P_G = \frac{x_{G,R} + x_{G,D} + x_{G,I}}{n}\) \(P_B = \frac{x_{B,R} + x_{B,D} + x_{B,I}}{n}\) \(P_U = \frac{x_{U,R} + x_{U,D} + x_{U,I}}{n}\) Next, calculate the expected proportions for each group using these overall proportions: \(E_{G,R} = n_R \cdot P_G\) \(E_{G,D} = n_D \cdot P_G\) \(E_{G,I} = n_I \cdot P_G\) \(E_{B,R} = n_R \cdot P_B\) \(E_{B,D} = n_D \cdot P_B\) \(E_{B,I} = n_I \cdot P_B\) \(E_{U,R} = n_R \cdot P_U\) \(E_{U,D} = n_D \cdot P_U\) \(E_{U,I} = n_I \cdot P_U\)

Perform a Chi-Square Test

To determine if there are significant differences in the proportions among the three groups, perform a chi-square test using the observed and expected counts for each affiliation. Calculate the test statistic, \(\chi^2\), using the following formula: \(\chi^2 = \sum \frac{(O - E)^2}{E}\) where \(O\) is the observed count and \(E\) is the expected count.

Determine the P-value and Make a Decision

Compare the calculated \(\chi^2\) statistic to the critical value from the chi-square distribution table. Since we have 2 degrees of freedom (since there are 3 groups - 1) and using a significance level of \(\alpha = 0.05\), we can find the critical value in the table. If the calculated \(\chi^2\) value is greater than the critical value, we reject the null hypothesis in favor of the alternative hypothesis, indicating a significant difference in the proportions among the three groups.

Calculate the Proportions for Each Affiliation (if applicable)

If there is a significant difference found in step 4, calculate the proportions for each group: For Republicans: \(p_{G,R} = \frac{x_{G,R}}{n_R}\) \(p_{B,R} = \frac{x_{B,R}}{n_R}\) \(p_{U,R} = \frac{x_{U,R}}{n_R}\) For Democrats: \(p_{G,D} = \frac{x_{G,D}}{n_D}\) \(p_{B,D} = \frac{x_{B,D}}{n_D}\) \(p_{U,D} = \frac{x_{U,D}}{n_D}\) For Independents: \(p_{G,I} = \frac{x_{G,I}}{n_I}\) \(p_{B,I} = \frac{x_{B,I}}{n_I}\) \(p_{U,I} = \frac{x_{U,I}}{n_I}\) These proportions can help us describe the nature of the differences between the opinions of Republicans, Democrats, and Independents on the Health Care Reform Act.

Key concepts

Hypothesis Testing

Hypothesis testing is a statistical method that helps us make decisions based on data. In this problem, we are trying to determine if there are significant differences in opinions about the Health Care Reform Act among different political affiliations: Republicans, Democrats, and Independents.

The process begins with stating two hypotheses. The **null hypothesis** ( H_0 ) assumes that there is no difference in opinions among these groups. In contrast, the **alternative hypothesis** ( H_a ) posits that at least one group's opinion differs. Hypothesis testing gives us a way to determine which hypothesis is more likely based on our data.

To decide if differences are significant, we use statistical tests like the chi-square test. The test compares observed data to expected data under the null hypothesis. If the test shows significant differences, we reject the null hypothesis and accept the alternative. This indicates that political affiliation may affect opinions about the law.

Proportions

Proportions are crucial for understanding data distributions, especially in survey results like this one. A **proportion** represents a part of a whole, calculated by dividing the count of interest by the total number of observations.

In our context, proportions help us compare how many people from each political group (Republicans, Democrats, Independents) think the health care law is good, bad, or are unsure. We first find the overall proportions, summarizing the entire survey, then use these proportions to predict what we'd expect to see in each political group if there were no differences.

For example, if 60% of all voters think the law is good, we would expect this trend to apply across all political affiliations. If our observed data differ substantially from what we expect, this suggests varying opinions among the groups. Calculating these proportions allows us to precisely describe how each group's views may differ.

Political Affiliation

Political affiliation can often influence individuals' opinions and perceptions, especially regarding policy matters. In this exercise, we're interested in whether Republicans, Democrats, and Independents differ in their views on the Health Care Reform Act.

By analyzing the data from different affiliations, we can understand broader patterns of political belief. For example, we might find that one group overwhelmingly supports the law while another is largely opposed. This insight is valuable because it not only reflects voters' current opinions but can also influence how parties might shape their future policies to align with or counteract public sentiment.

Understanding affiliation-based differences requires careful data handling and analysis, but it ultimately offers a clearer picture of the political landscape. Such insights are vital for policymakers, campaign strategists, and voters, providing a deeper understanding of how political identity impacts policy opinions.