"Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.
Tax Fraud In a survey of 1040 U.S. adults, 62 have had someone impersonate them to try to claim tax refunds. (Adapted from Pew Research Center)"

Verified step by step guidance

1. Identify the given information: The total number of U.S. adults surveyed is 1040, and the number of adults who have had someone impersonate them to claim tax refunds is 62.
2. Understand the concept: The point estimate for the population proportion (p̂) is calculated as the ratio of the number of successes (in this case, adults impersonated) to the total sample size. The formula is p̂ = x / n, where x is the number of successes and n is the total sample size.
3. Substitute the given values into the formula for p̂: Use x = 62 and n = 1040 to calculate p̂.
4. Find q̂, the complement of p̂: Recall that q̂ = 1 - p̂. Once p̂ is calculated, subtract it from 1 to find q̂.
5. Interpret the results: p̂ represents the proportion of adults in the sample who have been impersonated for tax fraud, and q̂ represents the proportion of adults in the sample who have not been impersonated for tax fraud.

Key Concepts

Population Proportion

The population proportion, denoted as p, represents the fraction of a population that possesses a certain characteristic. In this context, it refers to the proportion of U.S. adults who have experienced impersonation for tax fraud. Understanding this concept is crucial for estimating how widespread the issue is within the entire population based on survey data.

Point Estimate

A point estimate is a single value derived from sample data that serves as a best guess for a population parameter. In this case, p^ (p-hat) is the point estimate of the population proportion p, calculated by dividing the number of individuals with the characteristic (62) by the total sample size (1040). This estimate provides a quick snapshot of the population's behavior regarding tax fraud.

Complement of the Population Proportion

The complement of the population proportion, denoted as q, represents the proportion of the population that does not have the characteristic in question. It is calculated as q = 1 - p. In the context of the survey, q would indicate the proportion of U.S. adults who have not experienced impersonation for tax fraud, providing a complete view of the situation.