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An article in the Chicago Tribune (August \(29 .\) 999) reported that in a poll of residents of the Chicago suburbs, \(43 \%\) felt that their fina based on interviews with 930 randomly selected suburban residents. The sample included suburban Cook County plus DuPagc, Kanc, Lake, McHenry, and Will Countics. In a sample of this size, one can say with \(95 \%\) certainty that results will differ by no more than \(3 \%\) t from results obtained if all residents had been included in the poll." Comment on this statement. Give a statistical argument to justify the claim that the estimate of \(43 \%\) is within 396 of the true proportion of residents who feel that their financial situation has improved.ncial situation had improved during the past year. The following statement is from the article: "The findings of this Tribune poll are

Step by step solution

01

Understand Margin of Error

In a poll like this, the 'margin of error' is a statistic that quantifies the level of confidence that the pollsters have that the poll's results reflect the views of the entire population. In this case, it's mentioned that one can say with 95% certainty that the results differ by no more than 3% from the results that would be obtained if all residents had been included in the poll. This means the margin of error is 3%.

02

Calculate Confidence Interval

To validate the claim, we create a confidence interval using the given sample proportion (43%) and the margin of error (3%). The confidence interval is calculated as follows: Lower limit = Sample proportion - Margin of error = \(43% - 3% = 40%\) and Upper limit = Sample proportion + Margin of error = \(43% + 3% = 46% \). Therefore, the confidence interval is from 40% to 46%.

03

Interpretation

The statement from the Tribune means that if one conducted the same poll over and over again, 95% of the time the result would be in the range of 40% to 46%.

04

Evaluate the Poll Statement

Given the computed confidence interval, the claim that the estimate of 43% is within ±3% of the true proportion is justified statistically. It falls within the confidence interval that we computed.

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Key Concepts

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

Confidence Interval

In simple terms, a confidence interval is a range we use to express where we believe the true value lies. Imagine trying to figure out how many people in your neighborhood enjoy a particular hobby but you can only ask a handful of people due to time constraints. You might not be able to ask everyone, but you can provide an estimate based on those you did ask. The confidence interval helps to express this estimate with a measure of certainty.

Here's how it works:

• You use data from your sample to estimate the true proportion for the entire population.
• The interval is based on your selected confidence level, which represents how sure you are that the true value lies within that interval.
• In the context of the Tribune poll, a 95% confidence level means we're saying, "If we conducted the same poll many times, 95% of those polls would result in our true value falling within the interval of 40% to 46%."

The main takeaway is that confidence intervals give us a smart way to predict where the true population parameter falls, based on a random sample and its statistical certainty.

Sample Proportion

The sample proportion is like taking a small scoop of your favorite ice cream to see if the whole tub tastes good. It's a way to get an idea of the bigger picture from a smaller part of it. In surveys and polls, the sample proportion tells us something about the entire population but gathered from a smaller group.

Here's the breakdown:

• The sample proportion is calculated by dividing the number of favorable responses by the total number of respondents.
• In the case of the Chicago Tribune poll, the sample proportion was 43%, meaning that out of the 930 suburban residents surveyed, 43% felt that their financial situation had improved.
• The sample proportion serves as an estimate of the true population proportion and is used to calculate the confidence interval.

The concept of sample proportion is critical as it allows us to gauge broader public opinion or behavior efficiently without needing to survey or assess every single individual in the population.

Statistical Argument

A statistical argument is essentially making a case based on data. Just like a lawyer presents evidence and logic to support their claim, a statistical argument uses numbers and statistical tools to explain or justify a point.

To understand this better, let's dissect the claim from the Chicago Tribune poll:

• The poll reported a sample proportion of 43%, with a margin of error of ±3% and a 95% confidence level.
• The statistical argument here is that, given these parameters, the true proportion of residents who believe their financial situation improved falls within a range of 40% to 46%.
• Using statistical methods, such as calculating the confidence interval, strengthens the argument by providing a clear, quantifiable measure of reliability.
• This bolsters confidence in the poll's results and helps ensure informed conclusions are drawn.

Overall, a statistical argument helps substantiate claims with a robust and credible backing, making statistical evidence clear and reliable.