8. Sampling Distributions & Confidence Intervals: Proportion

In the context of the sampling distribution of the sample proportion $p$, why is a sample typically used instead of collecting data from the entire population?

Which of the following is not a property of the sampling distribution of the sample proportion?

Suppose a simple random sample of 1200 adults is selected from a large population in which the proportion of adults who support a certain policy is $p$. What is the mean of the sampling distribution of the sample proportion $\bar{p}$?

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.

Sampling Distribution of the Sample Standard Deviation For the following, round results to three decimal places.

a. Find the value of the population standard deviation σ.

Reincarnation Suppose 21% of all American teens (age 13–17 years) believe in reincarnation.

d. In a survey of 100 American teens, how many would you expect to believe in reincarnation?

Why is a sample used more often than a population when estimating a population proportion $p$?

In the context of sampling distributions of sample proportions, $\mathrm{population}\mathrm{parameters}$ are difficult to calculate due to which of the following reasons?

Which of the following statements about sampling distributions for the sample mean is false?

The probability of someone voting for a particular candidate in a two-person election is $45\%$ . Use a normal distribution to approximate the probability that between $62$ and $70$ people out of a sample of $100$ vote for the candidate. 

A previous study found that $80\%$ of people preferred drinking Pepsi over Coca Cola. Use a normal distribution to approximate the probability that a random sample of $100$ people reveals $60$ people or more preferring Pepsi. 

A previous study found that $80\%$ of people preferred drinking Pepsi over Coca Cola. Use a normal distribution to approximate the probability that, from this same random sample of $100$ people, that between $10$ and $11$ people prefer Coca Cola.

Fatal Car Crashes There are about 15,000 car crashes each day in the United States, and the proportion of car crashes that are fatal is 0.00559 (based on data from the National Highway Traffic Safety Administration). Assume that each day, 1000 car crashes are randomly selected and the proportion of fatal car crashes is recorded.

a. What do you know about the mean of the sample proportions?

College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample.

a. What is the approximate shape of the distribution of the sample means (uniform, normal, skewed, other)?

College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample.

b. What value do the sample means target? That is, what is the mean of all such sample means?

In Exercises 7–10, use the same population of {4, 5, 9} that was used in Examples 2 and 5. As in Examples 2 and 5, assume that samples of size n = 2 are randomly selected with replacement.

Sampling Distribution of the Sample Variance

a. Find the value of the population variance σ².