You Explain It! Valentine’s Day A Rasmussen Reports national survey of l000 adult Americans found that 18% dreaded Valentine’s Day. The margin of error for the survey was 4.5 percentage points with 95% confidence. Explain what this means.

Verified step by step guidance

1. Identify the sample proportion \( \hat{p} = 0.18 \) (18%) who dreaded Valentine's Day from the survey of 1000 adults.
2. Understand that the margin of error (MOE) is \( 4.5\% \), which means the true population proportion is estimated to lie within \( \hat{p} \pm \text{MOE} \).
3. Recognize that the confidence level is 95%, indicating that if we repeated this survey many times, approximately 95% of the confidence intervals constructed would contain the true population proportion.
4. Express the confidence interval as \( 0.18 \pm 0.045 \), or equivalently, from \( 0.135 \) to \( 0.225 \), meaning we are 95% confident the true proportion of adults who dread Valentine's Day is between 13.5% and 22.5%.
5. Explain that this interpretation relies on the assumption that the sample was randomly selected and that the sampling distribution of the sample proportion is approximately normal due to the large sample size.

Key Concepts

Margin of Error

The margin of error quantifies the range within which the true population parameter is expected to lie, based on the sample data. In this case, it means the actual percentage of adults who dread Valentine’s Day is likely within 4.5 percentage points above or below the reported 18%.

Confidence Level

The confidence level indicates the degree of certainty in the estimate, here 95%, meaning that if the survey were repeated many times, 95% of the calculated intervals would contain the true population value. It reflects the reliability of the survey results.

Sample Survey and Population Inference

A sample survey collects data from a subset of the population to make inferences about the whole group. The 1,000 adults surveyed represent the larger population of American adults, allowing statisticians to estimate attitudes toward Valentine’s Day with some uncertainty.