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Multiple choice: Select the best answer for Exercises 49 to 52. A Gallup Poll found that only 28% of American adults expect to inherit money or valuable possessions from a relative. The poll's margin of error was \(\pm 3\) percentage points at a 95% confidence level. This means that (a) the poll used a method that gets an answer within 3% of the truth about the population 95% of the time. (b) the percent of all adults who expect an inheritance is between 25% and 31%. (c) if Gallup takes another poll on this issue, the results of the second poll will lie between 25% and 31%. (d) there’s a 95% chance that the percent of all adults who expect an inheritance is between 25% and 31%. (e) Gallup can be 95% confident that between 25% and 31% of the sample expect an inheritance.

Step by step solution

01

Understand the Margin of Error

The margin of error given is \( \pm 3 \% \). This means that the actual percentage of American adults who expect to inherit money is likely within 3% of the sample result, which was 28%.

02

Calculate the Confidence Interval

Find the confidence interval by adding and subtracting the margin of error from the sample percentage. Thus, the lower bound is \( 28\% - 3\% = 25\% \) and the upper bound is \( 28\% + 3\% = 31\% \). Therefore, the interval is 25% to 31%.

03

Interpret the 95% Confidence Level

A 95% confidence level means that if we were to take many samples, about 95% of these samples would produce intervals that contain the true population parameter. This does not mean that there is a 95% probability for the specific interval calculated from our sample.

04

Evaluate Each Option

Option (a) is correct as it describes the method's long-term accuracy. Option (b) suggests a straightforward interpretation of the interval, but doesn't mention repeated sampling, so it's incorrect. Option (c) assumes future poll results fall into the interval, which is incorrect. Option (d) incorrectly states the probability of the parameter within the interval. Option (e) misinterprets the confidence level relating to the sample rather than the population.

05

Choose the Best Answer

The best answer is (a) because it correctly addresses the usage of the margin of error and confidence level in the context of polling. It indicates the methodology is set to be within 3% of the true value for 95% of similar samples.

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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 (MoE) is a critical component when interpreting poll results like those conducted by Gallup. Simply put, the MoE indicates how close the sample's percentage is to the true percentage of the population. In the Gallup Poll about inheritances, the margin of error is \( \pm 3 \% \). This signifies that the true percentage of American adults expecting an inheritance is likely within 3% of the survey result, which was 28%.
To understand this further:

• Lower bound: 28% - 3% = 25%
• Upper bound: 28% + 3% = 31%

Thus, the real percentage could realistically be anywhere from 25% to 31%. It's a range that gives insight into potential variation and the inherent uncertainty in sampling surveys.

95% Confidence Level

A 95% confidence level helps us understand the reliability of the poll results. This isn't about a single survey but is more about the process. If Gallup repeatedly took samples in the same manner many times, approximately 95% of these samples' results would produce a range (or confidence interval) containing the true rate of people expecting an inheritance.
It’s crucial to remember:

• Confidence doesn't mean certainty about a single survey.
• It’s about the likelihood that the method used captures the true population parameter.

Rather than thinking there's a 95% chance this specific range (25% to 31%) contains the true value, understand it as the method being highly reliable and trustworthy.

Statistical Sampling

Statistical sampling is the backbone of any polling process, including those by Gallup. It involves selecting a group (sample) from a larger population to draw conclusions about the entire group.

Key Points in Sampling:

• Samples should be random to avoid bias.
• The sample size needs to be large enough to be representative.

In Gallup's methodology, this sampling is carried out such that the sample represents the views of the broader population. Thus, conclusions about American adults, in general, can be reliably inferred from the sample, alongside the confidence intervals and margin of error calculations.

Gallup Poll

The Gallup Poll is a well-known source for gauging public opinion across various issues. It uses scientific methods for data collection, ensuring that results reflect the broader population's views as accurately as possible. In conducting polls:

• Gallup employs random sampling techniques to get unbiased data.
• They calculate margins of error and confidence intervals, which give a statistical sense of accuracy and reliability.
• Results help inform public discussion and policy decisions.

The discussed poll about inheritance reflects these robust principles, aiming to provide a clear picture of expectations around wealth transfer among American adults.