Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.
In a survey of 880 unmarried U.S. adults who are living with a partner, 73% say love was a major reason why they decided to move in together. The survey’s margin of error is ±4.8%. (Source: Pew Research Center)

검증된 단계별 안내

Step 1: Understand the problem. The problem provides a sample proportion (73%), a sample size (880), and a margin of error (±4.8%). The goal is to translate this information into a confidence interval and approximate the confidence level.

Step 2: Write the formula for a confidence interval for a population proportion. The general formula is:

p̂ ± E

, where

p̂

is the sample proportion and

E

is the margin of error.

Step 3: Substitute the given values into the formula. Here,

p̂ = 0.73

(73% as a decimal) and

E = 0.048

(4.8% as a decimal). The confidence interval is:

0.73 ± 0.048

Step 4: Calculate the lower and upper bounds of the confidence interval. The lower bound is

0.73 - 0.048

, and the upper bound is

0.73 + 0.048

. These bounds represent the range of plausible values for the population proportion.

Step 5: Approximate the confidence level. The margin of error (±4.8%) suggests a typical confidence level of 95%, as this is a common level used in surveys. However, the exact confidence level would depend on the critical value (z*) used in the calculation, which is not explicitly provided in the problem.

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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 73% is the sample proportion and the margin of error is ±4.8%, the confidence interval would be from 68.2% to 77.8%.

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 fall, based on the sample data. In this case, a margin of error of ±4.8% means that the true percentage of unmarried U.S. adults who believe love was a major reason for cohabitation could vary by this amount from the reported 73%.

Level of Confidence

The level of confidence reflects the degree of certainty that the confidence interval contains the true population parameter. Common levels of confidence are 90%, 95%, and 99%, with higher levels indicating greater certainty but wider intervals. The level of confidence can be approximated based on the sample size and the margin of error, often using standard normal distribution values.