91Ó°ÊÓ

On June 28,2012 the U.S. Supreme Court upheld the much debated 2010 healthcare law, declaring it constitutional. A Gallup poll released the day after this decision indicates that \(46 \%\) of 1,012 Americans agree with this decision. At a \(95 \%\) confidence level, this sample has a \(3 \%\) margin of error. Based on this information, determine if the following statements are true or false, and explain your reasoning. (a) We are \(95 \%\) confident that between \(43 \%\) and \(49 \%\) of Americans in this sample support the decision of the U.S. Supreme Court on the 2010 healthcare law. (b) We are \(95 \%\) confident that between \(43 \%\) and \(49 \%\) of Americans support the decision of the U.S. Supreme Court on the 2010 healthcare law. (c) If we considered many random samples of 1,012 Americans, and we calculated the sample proportions of those who support the decision of the U.S. Supreme Court, \(95 \%\) of those sample proportions will be between \(43 \%\) and \(49 \%\). (d) The margin of error at a \(90 \%\) confidence level would be higher than \(3 \%\).

Step by step solution

01

Understand Margin of Error

The margin of error is a range that indicates where the true proportion of the population parameter may lie, given a particular confidence level. Here, the margin of error is stated as \(3\%\), meaning we are \(95\%\) confident that the true proportion of Americans who agree falls within \(46\% \pm 3\%\). This results in a confidence interval of \(43\%\) to \(49\%\).

02

Evaluate Statement (a)

Statement (a) claims that the interval \(43\%\) to \(49\%\) reflects the opinions of Americans in the sample. This is incorrect because the confidence interval relates to the entire population, not just the sample. The interval tells us about the sample estimate's ability to reflect the population value, not about the sample itself.

03

Evaluate Statement (b)

Statement (b) states a confidence interval for the entire population of Americans, which is correct. The interval \(43\%\) to \(49\%\) is where we are \(95\%\) confident that the true proportion of the entire U.S. population's support lies, based on the sample data.

04

Evaluate Statement (c)

Statement (c) involves understanding the interpretation of a confidence interval over many samples. The claim is technically accurate: about \(95\%\) of intervals calculated from many samples of this size will contain the true population proportion. For any given interval (like \(43\%\) to \(49\%\)), it's not about proportions of various samples but about where the true proportion might be found.

05

Evaluate Statement (d)

Statement (d) involves understanding how changing confidence levels affects the margin of error. A \(90\%\) confidence level leads to a smaller margin of error because a lower confidence level typically corresponds to a narrower interval, not a broader one. Hence, this statement is false since the margin of error would actually be less than \(3\%\) for a \(90\%\) confidence level.

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Key Concepts

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

Margin of Error

The margin of error is a vital concept in statistics that helps us understand the accuracy of a survey's results. It provides a range that reflects how far off the sample's findings might be from the true population proportion. This margin shows us the potential error in our sample estimates due to random sampling.

In the context of the exercise, a margin of error of \(3\%\) at a \(95\%\) confidence level means we estimate the true proportion of Americans who agree with the Supreme Court decision to be between \(43\%\) and \(49\%\) (i.e., \(46\% \pm 3\%\)).

A smaller margin of error typically indicates a more precise estimate, assuming the confidence level remains constant. Understanding the margin of error helps us gauge how close our survey results are likely to reflect the true sentiments of the broader population.

Population Proportion

Population proportion refers to the fraction or percentage of an entire group that exhibits a particular characteristic. In our exercise, the population proportion is what we're trying to estimate—the actual percentage of all Americans who support the 2010 healthcare law.

When we survey a sample, like 1,012 Americans in this case, we use their responses to estimate the population proportion. However, we know it's impractical to survey everyone. Thus, the confidence interval (including the margin of error) helps us understand where the true population proportion likely falls, given our sample data.

It's important to remember that the sample proportion may not perfectly match the actual population proportion, but with adequate sample size and proper margin of error, we can make a very educated estimate.

Sample Data

Sample data is the information we gather from a subset of the population. In our exercise, the sample includes the opinions of 1,012 Americans regarding the Supreme Court's decision on healthcare. This small group represents the huge population of the United States.

We rely on sample data because it is typically impossible or impractical to survey an entire population. The accuracy and reliability of a sample in predicting population trends depend mostly on its size and random selection. A well-selected sample ensures that we can confidently extend its findings to the broader population.

Moreover, the margin of error and confidence level help us determine how much we can trust that our sample data reflects the entire population accurately.

Statistical Interpretation

Statistical interpretation involves making sense of the data through analysis and understanding the results' implications. It emphasizes correctly understanding what statistical statements mean.

In our exercise, statements about confidence intervals and margin of error require careful interpretation. For instance, statement (b) is true because it correctly identifies the confidence interval as it applies to the entire population, not merely the sample.
Statement (c) confirms that over numerous repeated samples, \(95\%\) of calculated confidence intervals will include the true population proportion.

Understanding these interpretations ensures that we use the statistics correctly and don't mislead ourselves or others with incorrect conclusions. It is crucial to grasp what confidence levels and intervals genuinely represent, as this directly influences the action or policy one might recommend based on these findings.