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. \(5 \varepsilon\) (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

Clarify the Margin of Error Statement

The margin of error is given as 3% at a 95% confidence level. This means if our sample proportion is 46%, we calculate the confidence interval as 46% ± 3%. This results in a confidence interval of 43% to 49%.

02

Evaluate Statement (a)

Statement (a) refers to the confidence interval for the sample. Since the margin of error applies to the population proportion, not the sample, statement (a) is false because the interval applies to the population proportion estimate.

03

Evaluate Statement (b)

Statement (b) claims that the confidence interval from 43% to 49% applies to the population proportion. This statement is true because the confidence interval provides an estimate range for the true population proportion.

04

Evaluate Statement (c)

Statement (c) describes the concept of multiple samples. At a 95% confidence level, if we repeatedly sample the population, 95% of the confidence intervals calculated from those samples would indeed contain the true population proportion. Thus, this statement is true as it aligns with the definition of a confidence interval.

05

Evaluate Statement (d)

Statement (d) concerns the margin of error for a 90% confidence level. A lower confidence level (90%) actually decreases the margin of error, so it would be less than 3%, not more. Therefore, statement (d) is false.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Margin of Error

The margin of error is an important concept when interpreting results from surveys and polls. In simple terms, the margin of error tells us the range in which the true population proportion is likely to fall, based on a sample proportion. For example, if a poll finds that 46% of people support a decision with a 3% margin of error, then the true support rate in the larger population is expected to be between 43% and 49%. This "buffer" accounts for sampling variability – the idea that different samples, even of the same size, might yield slightly different results.

A few key things to know about the margin of error:

• It's always attached to a confidence level (like 95%).
• A smaller margin of error indicates greater precision in the estimate.
• A larger sample size typically results in a smaller margin of error.
• The margin of error applies to the population parameter, not the sample.

Understanding this helps clarify the boundaries of our uncertainty when interpreting survey results.

Confidence Level

A confidence level indicates how likely it is that a given confidence interval contains the true population parameter. Commonly used confidence levels include 90%, 95%, and 99%, but 95% is often standard, as seen in the exercise from the Gallup poll.

Here’s what a 95% confidence level means:

• If the survey were repeated many times, 95% of the calculated confidence intervals would capture the true population proportion. This is a reflection of consistency rather than certainty for any one survey.
• Higher confidence levels mean more certainty but often come with a wider margin of error.
• To achieve a smaller margin of error with the same confidence level, a larger sample size would be needed.
• The confidence level does not describe the probability that any specific interval contains the parameter.

In essence, the confidence level helps us measure how confident we can be in the results of our sampling method without making a specific claim about the exact result for any particular interval.

Population Proportion

The population proportion is a fundamental concept in statistics that represents the proportion of a certain characteristic within a whole population. In surveys or polls, the population proportion might refer to the percentage of people who support a certain policy or prefer a particular product.

In the Gallup poll exercise, the population proportion we're interested in is the percentage of Americans who support the Supreme Court's decision on the healthcare law. The sample proportion obtained from the poll is 46%. By applying the margin of error and confidence interval, we estimate that the true population proportion of support likely falls between 43% and 49%.

Important points to understand about population proportion:

• It is an unknown parameter that we try to estimate using sample data.
• The accuracy of estimates for population proportions depends largely on sample size and sampling method.
• Different samples provide different estimates, which is why we use a margin of error to describe uncertainty.

Grasping this concept is vital for interpreting survey data correctly and understanding the typical variability that comes with sample-based research.