Finding p^ and q^ In Exercises 3–6, let p be the population proportion for the situation. Find point estimates of p and q.
Social Security In a survey of 661 non-retired Americans, 218 said that they expect to rely on Social Security as major source of income when they retire. (Adapted from Gallup)

검증된 단계별 안내

Step 1: Understand the problem. The goal is to find the point estimates of p (the sample proportion of individuals who expect to rely on Social Security as a major source of income) and q (the complement of p, representing those who do not expect to rely on Social Security).

Step 2: Recall the formula for the sample proportion p̂. It is given by:

p^= \frac{x}{n}

, where x is the number of individuals with the characteristic of interest (218 in this case), and n is the total sample size (661 in this case).

Step 3: Substitute the given values into the formula for p̂. This will give you the point estimate for p̂:

p^= \frac{218}{661}

Step 4: Recall that q̂ is the complement of p̂, which can be calculated using the formula:

q^= 1 - p^

. Use the value of p̂ from Step 3 to compute q̂.

Step 5: Interpret the results. The values of p̂ and q̂ represent the proportions of the sample that expect to rely on Social Security and do not expect to rely on Social Security, respectively.

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주요 개념

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

Population Proportion

The population proportion, denoted as p, is the ratio of members of a population that have a particular characteristic to the total number of members in that population. In this context, it represents the proportion of non-retired Americans who expect to rely on Social Security as a major source of income when they retire.

Point Estimate

A point estimate is a single value that serves as an approximation of a population parameter. In this case, p^ (p-hat) is the point estimate of the population proportion p, calculated from sample data. It provides a quick summary of the sample's characteristics, allowing for easier interpretation of the data.

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 of interest. It is calculated as q = 1 - p. In this scenario, q^ (q-hat) would indicate the proportion of non-retired Americans who do not expect to rely on Social Security as a major source of income.