91Ó°ÊÓ

A November 2011 Gallup poll asked American adults about their views of healthcare and the healthcare system in the United States. Although feelings about the quality of healthcare were positive, the same cannot be said about the quality of the healthcare system. According to this study, \(29 \%\) of Independents and \(27 \%\) of Democrats rated the healthcare system as being excellent or good (www.gallup.com/poll/ \(150788 /\) Americans-Maintain- Negative-View-Healthcare-Coverage.aspx). Suppose that these results were based on samples of 1200 Independents and 1300 Democrats. a. Let \(p_{1}\) and \(p_{2}\) be the proportions of all Independents and all Democrats, respectively, who will rate the healthcare system as being excellent or good. Construct a \(97 \%\) confidence interval for \(p_{1}-p_{2}\) b. Using a \(1 \%\) significance level, can you conclude that \(p_{1}\) is different from \(p_{2}\) ? Use both the critical-value and the \(p\) -value approaches.

Step by step solution

01

Find Proportions

From the problem, we have proportions of \(29\%\) Independents and \(27\%\) Democrats who rate the healthcare system as being excellent or good, from the samples of 1200 Independents and 1300 Democrats which is \(0.29\) and \(0.27\), respectively.

02

Construct Confidence Interval for \(p_{1}-p_{2}\)

A confidence interval for the difference in proportions is given by \((p_{1} - p_{2}) \pm Z_{\alpha/2} * \sqrt{ \left(\frac{p_{1}(1-p_{1})}{n_{1}}\right) + \left(\frac{p_{2}(1-p_{2})}{n_{2}}\right)}\), where \(Z_{\alpha/2}\) is the Z score from the standard normal distribution corresponding to \(\alpha/2\). For a \(97\%\) confidence interval, \(\alpha = 0.03\) and \(Z_{\alpha/2} = 2.17\). Substituting the given values, we can calculate the interval.

03

Hypothesis Testing

We can test the hypothesis \(H_{0} : p_{1} = p_{2}\) against \(H_{a} : p_{1} \neq p_{2}\) using a Z test. The test statistic is calculated as \(Z = \frac{(p_{1} - p_{2}) - 0}{\sqrt{\left(\frac{p_{1}(1-p_{1})}{n_{1}}\right) + \left(\frac{p_{2}(1-p_{2})}{n_{2}}\right)}}\). We then compare this value with the critical values calculated as \(\pm Z_{\alpha/2}\) and the P-value calculated from the standard normal distribution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Confidence Intervals

When we talk about confidence intervals, we're referring to a range of values used to estimate a population parameter. Here, the parameter in question is the difference between two proportions, specifically the ratings by Independents and Democrats. A confidence interval gives us a range where we expect the true difference to lie with a certain level of certainty. For example, a 97% confidence interval means there's a 97% chance the true difference falls within that interval.
To calculate this, we use the formula for the confidence interval of a difference in proportions. The formula is:

• \[(p_{1} - p_{2}) \pm Z_{\alpha/2} \times \sqrt{ \left(\frac{p_{1}(1-p_{1})}{n_{1}}\right) + \left(\frac{p_{2}(1-p_{2})}{n_{2}}\right)}\]

In this formula:

• \(p_1\) and \(p_2\) are sample proportions (0.29 and 0.27 respectively).
• \(n_1\) and \(n_2\) are the sample sizes (1200 and 1300).
• \(Z_{\alpha/2}\) is the Z-score corresponding to the desired confidence level (2.17 for 97%).

By substituting these values into the formula, we can compute the confidence interval, providing insight into how much the ratings might differ between the two groups.

Hypothesis Testing Explained

Hypothesis testing is designed to assess whether there's enough statistical evidence to support a specific claim about the population. In this exercise, we want to see if the proportion of Independents rating the healthcare system positively is different from that of Democrats. This involves:

• Null Hypothesis \(H_0: p_1 = p_2\) (Independents and Democrats have the same positive ratings).
• Alternative Hypothesis \(H_a: p_1 eq p_2\) (There is a difference in ratings).

To determine if there's a significant difference, we perform a Z test for the difference between the two proportions. The formula for the test statistic Z is:

• \[Z = \frac{(p_{1} - p_{2}) - 0}{\sqrt{\left(\frac{p_{1}(1-p_{1})}{n_{1}}\right) + \left(\frac{p_{2}(1-p_{2})}{n_{2}}\right)}}\]

This result is then compared to critical values from the standard normal distribution for the chosen significance level (1% in this case). If the calculated Z falls outside the range of these critical values, we reject the null hypothesis, suggesting a statistically significant difference.

Proportion Difference in Context

A difference in proportions allows us to compare two different groups or populations to understand how one might behave differently from the other. Here, the focus is on the perceived quality of the healthcare system by Independents versus Democrats. By evaluating the difference in their proportions, we gain insights into the variations in opinion.
When analyzing proportion differences, the key is understanding:

• The significance level (\(1\%\)) tells us how much risk we are willing to take for a false positive (saying there is a difference when there isn’t).
• Confidence Interval (\(97\%\)) provides the range where the true difference in opinions might lie.

Through these statistical tools, we not only identify if a difference exists, but how meaningful that difference might be, allowing for deeper insights into group behaviors and opinions.