You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% of the population proportion.
a. No preliminary estimate is available. Find the minimum sample size needed.

검증된 단계별 안내

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

n = \frac{z^{2}}{}

, where

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 = \frac{1}{2}

(or 0.5). This value maximizes the product

p (1 - p)

, ensuring the sample size is large enough.

Identify the z-score corresponding to a 95% confidence level. For a 95% confidence level, the z-score is approximately 1.96. This value is derived from the standard normal distribution.

Set the margin of error

E

to 0.05 (5%). This is the desired level of accuracy for the estimate.

Substitute the values into the formula:

n = \frac{{1.96}^{2}}{}

. Simplify the expression to calculate the minimum sample size needed.

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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. adults who have taken or planned to take a winter vacation. It is denoted by 'p' and is crucial for estimating how widespread a behavior or opinion is within a 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 results of a study are statistically valid. In this context, it involves using the desired confidence level and margin of error to find the minimum number of respondents required to accurately estimate the population proportion.

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 reflects the uncertainty associated with estimating the population proportion and is influenced by the sample size and variability in the data.