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A confidence interval provides a range of values that is likely to contain the population parameter with a certain level of confidence. In this context, we are trying to determine the percentage of Independents who oppose the National Health Plan.

To form a confidence interval, we first calculate the proportion of Independents who oppose the plan. Here, if 55% support, then 45% oppose it. The next step is to calculate the standard error (SE) using the formula:

\[SE = \sqrt{\frac{p(1-p)}{n}}\]
where \(p\) is the sample proportion (0.45) and \(n\) is the sample size (617).

The SE helps identify the range where the true proportion is likely to fall. By using 1.96 for a 95% confidence interval, the calculation
\[0.45 \pm 1.96 \times SE\]
gives us the interval \([0.411, 0.489]\). This indicates that the proportion of Independents who oppose the plan is between 41.1% and 48.9% with 95% confidence. Notice that 0.5 is not included, suggesting it is unlikely the true proportion who oppose is 50%.

Hypothesis testing is a statistical method that helps determine if there's enough evidence to support a specific claim about a population. Here, we want to test if a majority of Independents support the National Health Plan.

To do this, we perform a proportion hypothesis test.

The null hypothesis (\(H_0\)) states that the proportion of Independents supporting the plan is 0.5, meaning no majority. The alternative hypothesis (\(H_a\)) proposes that the proportion is greater than 0.5. First, we calculate the observed proportion from the sample:\[p = \frac{339}{617} \approx 0.5491\]
The next step is to calculate the test statistic, which evaluates how far our sample proportion is from the null hypothesis:
\[Z = \frac{p - 0.5}{SE_{H_0}}\]
where \(SE_{H_0} = \sqrt{\frac{0.5 \times (1-0.5)}{617}}\).

A Z-score is then used to get a p-value, showing the probability of observing a sample proportion as extreme as 0.5491, assuming the null hypothesis is true. A very low p-value would indicate strong evidence against \(H_0\), suggesting that a majority do indeed support the plan.

Polling data analysis is crucial for understanding public opinion. In the given exercise, polling involves examining the support for a National Health Plan among different political groups. Such data analysis provides insights that can be applied to broader societal and political trends.

In this analysis, the raw data consists of proportions - observed percentages of support within Democrats, Republicans, and Independents. The polling results relay immediate information: 79% of Democrats, 55% of Independents, and only 24% of Republicans support the plan.

Analyzing the data accurately involves statistical techniques:

• Proportion calculations to derive meaningful conclusions about majorities within groups.
• Confidence intervals to estimate the extent of public opinion with quantifiable uncertainty.
• Hypothesis testing to validate claims (e.g., a majority of Independents supporting the policy).

Using these methods ensures that political analysts and policymakers make informed decisions based on robust statistical evidence, thereby guiding subsequent actions and discussions in public policy.