Question

An article in the Chicago Tribune (August 29, 1999) reported that in a poll of residents of the Chicago suburbs, \(43 \%\) felt that their financial situation had improved during the past year. The following statement is from the article: "The findings of this Tribune poll are based on interviews with 930 randomly selected suburban residents. The sample included suburban Cook County plus DuPage, Kane, Lake, McHenry, and Will Counties. In a sample of this size, one can say with \(95 \%\) certainty that results will differ by no more than \(3 \%\) from results obtained if all residents had been included in the poll." Give a statistical argument to justify the claim that the estimate of \(43 \%\) is within \(3 \%\) of the actual percentage of all residents who feel that their financial situation has improved.

Short answer

The statement that 43% is within 3% of the actual percentage is justified using confidence intervals. It means that we are 95% confident that the true population proportion lies within the interval \(43% \pm 3%\). Repeating this method, we would expect 95% of the calculated intervals to contain the true parameter.

Step-by-step solution

Recall the concept of Confidence Intervals

A Confidence Interval is a type of interval estimate of a population parameter. It provides an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data. In any situation, the estimated parameter value lies at the center of the confidence interval, and the margin of error constitutes the rest of the interval.

Understand the Statement

In the given problem, it is stated that the results from the random sample size (930 people) would differ by no more than 3% from results we would have obtained if all residents had been included in the poll. This means we've got a confidence interval. The estimate of 43% (the proportion of residents whose financial situation improved) is the center of the interval, and the difference of no more than 3% provides our margin of error. So, the confidence interval is: \(43% \pm 3%\).

Justification

Now, since the confidence level is 95%, it means that we would expect that 95% of the confidence intervals calculated from these random samples will contain the true population parameter. That is to say, if we were to sample again using the same method, we're 95% confident that the obtained proportion of residents whose financial situation improved would lie within this calculated interval. Thus, the 43% is justifiably within 3% of the actual percentage with 95% certainty.