Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.
In a survey of 1052 parents of children ages 8–14, 68% say they are willing to get a second or part-time job to pay for their children’s college education, and 42% say they lose sleep worrying about college costs. The survey’s margin of error is ±3%. (Source: T. Rowe Price Group, Inc.)

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Identify the key components of the problem: the sample proportion (p̂), the sample size (n), and the margin of error (E). Here, p̂ = 0.68 (68%), n = 1052, and E = 0.03 (3%).

Understand that a confidence interval is constructed as p̂ ± E, where p̂ is the sample proportion and E is the margin of error. This gives the range of plausible values for the population proportion.

Substitute the values into the formula for the confidence interval: [p̂ - E, p̂ + E]. Using the given values, the confidence interval becomes [0.68 - 0.03, 0.68 + 0.03].

Interpret the confidence interval: The interval represents the range of values within which the true population proportion is likely to fall, given the margin of error.

Approximate the level of confidence: The margin of error (±3%) suggests a typical confidence level of 95%, as this is commonly associated with such surveys unless otherwise specified.

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주요 개념

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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed as an interval estimate, typically calculated as the sample proportion plus or minus the margin of error. For example, if 68% of surveyed parents are willing to get a second job, a confidence interval might be calculated to show the range within which the true proportion of all parents falls.

Margin of Error

The margin of error quantifies the uncertainty associated with a sample estimate. It indicates the range within which the true population parameter is expected to lie, based on the sample data. In this case, a margin of error of ±3% means that the true percentage of parents willing to get a second job could be as low as 65% or as high as 71%.

Level of Confidence

The level of confidence reflects the degree of certainty that the population parameter lies within the confidence interval. Common levels of confidence are 90%, 95%, and 99%, with higher levels indicating greater certainty but wider intervals. In this context, the level of confidence can be approximated based on the sample size and margin of error, often using standard statistical tables.