Ch. 6 - Confidence Intervals

Larson - Elementary Statistics: Picturing the World 8th Edition

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Larson 8th Edition

Ch. 6 - Confidence Intervals

문제 6.3.17a

6장, 문제 6.3.17a

Congress You wish to estimate, with 95% confidence, the population proportion of likely U.S. voters who think Congress is doing a good or excellent job. Your estimate must be accurate within 4% of the population proportion.
a. No preliminary estimate is available. Find the minimum sample size needed.

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Determine the formula for the minimum sample size needed to estimate a population proportion with a given margin of error. The formula is:

n = (z² * p * (1 - p)) / E²

, where

n

is the sample size,

z

is the z-score corresponding to the confidence level,

p

is the estimated population proportion, and

E

is the margin of error.

Since no preliminary estimate of the population proportion is available, use

p = 0.5

. This value maximizes the product

p * (1 - p)

, ensuring the sample size is large enough.

Identify the z-score for a 95% confidence level. For a 95% confidence level, the z-score is approximately

1.96

Substitute the values into the formula:

n = (1.96² * 0.5 * (1 - 0.5)) / 0.04²

. Here,

E = 0.04

represents the margin of error (4%).

Simplify the expression step by step to calculate the minimum sample size

n

. Round up to the nearest whole number, as sample size must be an integer.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Population Proportion

The population proportion refers to the fraction of a population that possesses a certain characteristic, in this case, U.S. voters who believe Congress is performing well. It is denoted by 'p' and is crucial for estimating how representative a sample will be in reflecting the views of the entire population.

Sample Size Calculation

Sample size calculation is a statistical method used to determine the number of observations or replicates needed to ensure that the sample accurately reflects the population. For estimating proportions, the formula incorporates the desired confidence level, margin of error, and the estimated population proportion to derive the minimum sample size required.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence, such as 95%. It provides a measure of uncertainty around the estimate and is essential for understanding the precision of the sample proportion in relation to the true population proportion.