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The General Social Survey asked a random sample of 1,390 Americans the following question: "On the whole, do you think it should or should not be the government's responsibility to promote equality between men and women?" \(82 \%\) of the respondents said it "should be". At a \(95 \%\) confidence level, this sample has \(2 \%\) 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 \(80 \%\) and \(84 \%\) of Americans in this sample think it's the government's responsibility to promote equality between men and women. (b) We are \(95 \%\) confident that between \(80 \%\) and \(84 \%\) of all Americans think it's the government's responsibility to promote equality between men and women. (c) If we considered many random samples of 1,390 Americans, and we calculated \(95 \%\) confidence intervals for each, \(95 \%\) of these intervals would include the true population proportion of Americans who think it's the government's responsibility to promote equality between men and women. (d) In order to decrease the margin of error to \(1 \%\), we would need to quadruple (multiply by 4 ) the sample size. (e) Based on this confidence interval, there is sufficient evidence to conclude that a majority of Americans think it's the government's responsibility to promote equality between men and women.

Step by step solution

01

Understanding the Margin of Error

The margin of error for this survey is given as 2%, which means that we can add and subtract this value from the sample proportion to form the confidence interval. Therefore, for the sample proportion of 82%, the confidence interval would be from 80% to 84%.

02

Evaluating Statement (a)

Statement (a) refers to the confidence interval for the sample. Since the confidence interval (80% to 84%) is based on the sample, it is true that we are 95% confident that between 80% and 84% of the sample holds the opinion that the government should promote equality.

03

Evaluating Statement (b)

Statement (b) generalizes the sample confidence interval to the entire population. Confidence intervals are used to estimate the range in which the true population parameter lies. Hence, this statement is true; the confidence interval estimates the population proportion.

04

Evaluating Statement (c)

Statement (c) describes the interpretation of confidence intervals across multiple samples. By definition of confidence intervals, 95% of these intervals would contain the true population proportion when using a 95% confidence level. Hence, this statement is true.

05

Evaluating Statement (d)

To halve the margin of error from 2% to 1%, we need to consider the relationship between sample size and margin of error. The margin of error is inversely proportional to the square root of the sample size, hence reducing the margin of error by a factor of 2 requires quadrupling the sample size. Therefore, statement (d) is true.

06

Evaluating Statement (e)

The confidence interval from 80% to 84% lies entirely above 50%, indicating a majority. Therefore, there is sufficient evidence to conclude that a majority of Americans believe the government should promote equality, making statement (e) true.

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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 critical part of statistical analysis, particularly when making estimates about a population based on a sample. It reflects the degree of uncertainty or variability in our sample estimate. For example, when we say that the proportion of people supporting a certain idea is 82% with a 2% margin of error, we mean that the real proportion could reasonably be as low as 80% or as high as 84%. This range creates what we call a confidence interval.

Why does the margin of error matter? Because it provides a measure of how much sample estimates might differ from the actual population parameter. A smaller margin of error indicates more precision in our estimate, which is often tied to a larger sample size. Understanding this helps one grasp the conclusions that can be reliably drawn from the data.

Population Proportion

Population proportion refers to the fraction of a population that possesses a certain characteristic. In our scenario, 82% of the survey respondents believe that it is the government's responsibility to promote equality between men and women. This is a reflection of the sample proportion, which is an estimate of the actual population proportion.

When we create a confidence interval from our sample proportion, we are attempting to capture that true population proportion within a specified range. It's crucial to remember that the real population proportion is an unknown value, and the confidence interval is our best estimate based on the available data. The importance of this concept lies in its ability to help make inferences about the entire population based on sample data.

Sample Size

Sample size is fundamental in determining the reliability and accuracy of any statistical analysis. A larger sample size leads to more reliable estimates by reducing the margin of error. This happens because the variability in sample statistics, like the sample proportion, decreases as the sample size increases.

For instance, if we wish to decrease the margin of error from 2% to 1%, we would need to quadruple our sample size. This is because the margin of error is inversely proportional to the square root of the sample size. As you test more people, you're more likely to capture the true population proportion, helping you make more accurate statistical inferences.

95% Confidence Level

A 95% confidence level is a standard choice in statistics to express the degree of certainty that a true population parameter falls within the calculated confidence interval. In practical terms, it means that if we were to take 100 different samples and calculate a confidence interval from each, about 95 of those intervals would contain the true population proportion.

This concept is crucial for understanding statistical significance. It doesn't mean there's a 95% chance the true proportion is in the interval, but rather that 95% of similarly constructed intervals will capture the true parameter. This understanding helps clarify why decisions or conclusions can be made with a specified degree of confidence based on the sample data.