Question

The voters in the midterm 2006 elections showed that Democrat and Republican differences extend beyond matters of opinion, and actually include the way they see the world. \({ }^{12}\) The three most important issues mentioned by Democrats and Republicans are listed below. $$\begin{array}{llll} & \text { 1st } & \text { 2nd } & \text { 3rd } \\\\\hline \text { Republicans } & \text { Terrorism } & \text { Economy } & \text { lraq war } \\\\(n=995) & 42 \% & 41 \% & 37 \% \\\\\text { Democrats } & \text { Iraq war } & \text { Economy } & \text { Health care } \\\\(n=1094) & 60 \% & 44 \% & 44 \%\end{array}$$ Use a large-sample estimation procedure to compare the proportions of Republicans and Democrats who mentioned the economy as an important issue in the elections. Explain your conclusions.

Short answer

Based on the z-test for proportions, the confidence interval for the difference in proportions between Republicans and Democrats mentioning the economy as an important issue in the midterm 2006 elections is (-0.03, 0.01). Since this interval includes zero, we can conclude that there is no significant difference between the two groups at a 95% confidence level.

Step-by-step solution

Identify the sample proportions

In the given data, 41% of Republicans (n=995) and 44% of Democrats (n=1094) mentioned the economy as an important issue. We can represent these as sample proportions denoted as \(p_{1}\) and \(p_{2}\) for Republicans and Democrats respectively: $$p_1 = 0.41$$ $$p_2 = 0.44$$

Calculate the pooled proportion

The pooled proportion is the combined proportion of both groups mentioning the economy. We can calculate this value with the formula: $$p = \frac{p_{1}n_{1} + p_{2}n_{2}}{n_{1}+n_{2}}$$ Plugging in values: $$p = \frac{(0.41)(995) + (0.44)(1094)}{995+1094}$$ $$p \approx 0.4259$$

Calculate the z-score

The z-score, denoted as z, is computed using the following formula: $$z = \frac{(p_1 - p_2) - 0}{\sqrt{\frac{p(1-p)}{n_1} + \frac{p(1-p)}{n_2}}}$$ Plugging in the values: $$z = \frac{(0.41 - 0.44) - 0}{\sqrt{\frac{0.4259(1-0.4259)}{995} + \frac{0.4259(1-0.4259)}{1094}}}$$ $$z \approx -1.996$$

Determine the margin of error and confidence interval

At a 95% confidence level, the z-critical value is approximately 1.96 (from the z-distribution table). The margin of error can be calculated using the following formula: $$MOE = z_{\text{critical}} \times \sqrt{\frac{p(1-p)}{n_1} + \frac{p(1-p)}{n_2}}$$ Plugging in the values: $$MOE = 1.96 \times \sqrt{\frac{0.4259(1-0.4259)}{995} + \frac{0.4259(1-0.4259)}{1094}}$$ $$MOE \approx 0.029$$ Now, we calculate the 95% confidence interval (difference in proportions \(\pm\) margin of error): $$( -0.03, 0.01)$$

Draw conclusions

Since the confidence interval contains zero (0), we can conclude that there is no significant difference in proportions between Republicans and Democrats mentioning the economy as an important issue in the midterm 2006 elections at a 95% confidence level.

Key concepts

Confidence Interval

In statistics, a confidence interval is a range of values that's used to estimate the true value of an unknown population parameter.
A confidence interval is calculated from the statistics of the observed data, and provides a range of values that are likely to contain the parameter of interest.
For example, in this exercise, the confidence interval addresses the difference in proportions of Republicans and Democrats who consider the economy an important issue.
Typically, a 95% confidence interval is used, which implies that if we were to take 100 different samples and compute a confidence interval for each, we expect about 95 of those intervals to contain the true difference in proportions.
The formula for a simple confidence interval for the difference in proportions is:

• Difference in proportions \((p_1 - p_2)\) ± Margin of Error (MOE)

The MOE is derived from the critical value of the z-score, which quantifies the level of confidence, multiplied by the standard error of the difference.

Z-score

A Z-score is a statistical measurement that describes a value's relation to the mean of a group of values.
It tells us how many standard deviations an element is from the mean.
In hypothesis testing, a Z-score is used to determine whether to reject the null hypothesis.
For comparing proportions, like in our exercise, the Z-score helps us understand how different our observed sample proportions are from what we’d expect if there were no differences.
The formula used in this exercise is:

• \(z = \frac{(p_1 - p_2)}{\sqrt{\frac{p(1-p)}{n_1} + \frac{p(1-p)}{n_2}}}\)

The numerator \(p_1 - p_2\) is the difference between the two sample proportions, and the denominator is the standard error derived from the pooled estimate of the proportion \(p\).
A Z-score near 0 suggests no significant difference, while a larger or smaller Z-score might indicate a statistically significant difference.

Proportions Comparison

Proportions comparison is about evaluating the differences between two ratios or percentages from independent samples.
For example, this exercise compared the proportion of Republicans and Democrats citing the economy as an important issue.
We can use statistical tests to ascertain if observed differences could occur by chance or if they are statistically significant.
By using proportions comparison, we compute a metric like the difference \(p_1 - p_2\) and then analyze this difference through confidence intervals and hypothesis tests.
If the confidence interval for the difference doesn't contain zero, we conclude that there is a statistically significant difference between the proportions.
In this exercise, the confidence interval of \((-0.03, 0.01)\) contains zero, implying no significant difference.

Large-sample Estimation

Large-sample estimation refers to approximations used when the sample size is sufficiently large.
In statistical analysis, larger samples provide more reliable estimations of population parameters because the sample better represents the population.
This exercise uses large-sample methods to compare the proportions of Republicans and Democrats regarding economic issues.
The formulae for pooled proportions, standard errors, and resultant Z-scores rely on large sample approximations to the normal distribution.
For two groups with sizes \(n_1 = 995\) and \(n_2 = 1094\), it's considered a large enough sample for the Central Limit Theorem to apply.
This makes the Z-test a valid approximation for testing differences in proportions.

• Pooled proportion: \(p = \frac{p_{1}n_{1} + p_{2}n_{2}}{n_{1}+n_{2}}\)

These methods rely heavily on having enough observations to ensure the normal distribution of the sample mean.