Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

문제 6.3.29

6장, 문제 6.3.29

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.
In a survey of 1502 U.S. adults, 31% said that they use Pinterest. The survey’s margin of error is ±2.9%. (Source: Pew Research Center)

검증된 단계별 안내

Step 1: Understand the problem. The problem provides a sample proportion (31%), a sample size (1502), and a margin of error (±2.9%). 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 form is:

p̂ ± ME

, where

p̂

is the sample proportion and ME is the margin of error.

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

p̂ = 0.31

(31% expressed as a decimal) and

ME = 0.029

(2.9% expressed as a decimal). The confidence interval is:

0.31 ± 0.029

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

0.31 - 0.029

, and the upper bound is

0.31 + 0.029

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

Step 5: Approximate the confidence level. The margin of error is typically associated with a 95% confidence level unless otherwise stated. Therefore, the confidence interval is likely based on a 95% confidence level.

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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 statistic plus or minus a margin of error. For example, if 31% of surveyed adults use Pinterest with a margin of error of ±2.9%, the confidence interval would be from 28.1% to 33.9%.

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 ±2.9% means that the true percentage of U.S. adults using Pinterest could be 2.9% higher or lower than the reported 31%.

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 margin of error, often using standard normal distribution values.

Translating Statements In Exercises 29–34, translate the statement into a confidence interval. Approximate the level of confidence.In a survey of 1502 U.S. adults, 31% said that they use Pinterest. The survey’s margin of error is ±2.9%. (Source: Pew Research Center)