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Payments In a May 2007 Experian/Gallup Personal Credit Index poll of 1008 U.S. adults aged 18 and over, \(8 \%\) of respondents said they were very uncomfortable with their ability to make their monthly payments on their current debt during the next three months. A more detailed poll surveyed 1288 adults, reporting similar overall results and also noting differences among four age groups: \(18-29,30-49,50-64,\) and \(65+.\) a) Do you expect the \(95 \%\) confidence interval for the true proportion of all 18 - to 29 -year-olds who are worried to be wider or narrower than the \(95 \%\) confidence interval for the true proportion of all U.S. consumers? Explain. b) Do you expect this second poll's overall margin of error to be larger or smaller than the Experian/Gallup poll's? Explain.

Step by step solution

01

Understanding Confidence Intervals

A confidence interval gives a range of values that is likely to contain the true value of a population parameter. A larger sample size generally results in a narrower confidence interval because it provides more data points, leading to more precision.

02

Analyze Sample Size and Confidence Interval Width (Part a)

The first poll surveyed 1008 adults, while the detailed poll focused on subsets of 1288 adults divided by age. The detailed poll's sample for the 18-29 age group will be smaller than 1008. A smaller sample size increases variability, resulting in a wider confidence interval.

03

Compare Overall Sample Sizes (Part b)

The second poll's total sample size is 1288, larger than 1008 from the initial poll, which suggests a smaller margin of error for the overall population estimate in the second poll compared to the first.

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Key Concepts

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

Sample Size

The sample size refers to the number of individuals surveyed or tested in a poll or experiment. It's crucial because it impacts the reliability of the results. The general rule in statistical analysis is that larger sample sizes offer more reliable data. This occurs because:

• A larger sample represents a population more accurately.
• It tends to normalize the extremes (i.e., outliers) within a group.
• Data from more individuals reduce the impact of random errors.

In the context of the mentioned polls, the original survey had 1008 respondents, while the more detailed survey included 1288 respondents. Thus, when comparing these two, the larger sample size (1288) typically yields a more precise estimate, assuming all other factors are constant. This leads directly into the next concept, the margin of error, because one reason for a smaller margin of error is an increased sample size.

Margin of Error

The margin of error is a statistic that expresses the amount of random sampling error in a survey's results. It represents the range within which the true population parameter is expected to fall, based on the sampled data. A smaller margin of error indicates more confidence in the results of the poll. Here’s why:

• A larger sample size reduces the margin of error, as seen in the second poll with 1288 adults.
• It quantifies uncertainty by showing the extent to which the sample’s estimate might differ from the true population parameter.

For the second poll, which has a larger sample size than the first, we expect a smaller margin of error. This is because the increased number of respondents provides a more comprehensive view of the population, decreasing variability in the results.

Population Parameter

A population parameter is a value that gives information about a population. This might be a mean, median, percentage, or other statistical measure that summarizes certain aspects of the population.

• Unlike the sample statistic, which describes the sample, the population parameter describes the entire population.
• For surveys, the true population parameter is what researchers aim to estimate using confidence intervals and sample statistics.

In the context of the polls discussed, we are trying to estimate the proportion of the entire population (e.g., all U.S. adults) who feel uncomfortable about making payments. The confidence interval derived from the sample provides an estimated range for this population parameter, acknowledging that it might not be a perfectly accurate match due to sample limitations.