Unit 6: Confidence Intervals for Proportions Review

The manager of a city recreation center wants to estimate the percent of city residents who favor a proposal to build a new dog park. To gather data, the manager will select a random sample of city residents. Which of the following is the most appropriate interval for the manager to use for such an estimate?

• A one-sample z-interval for a sample proportion

• A one-sample z-interval for a population proportion

• A one-sample z-interval for a difference between population proportions

• A two-sample z-interval for a difference between sample proportions

• A two-sample z-interval for a difference between population proportions

Biologists studying horseshoe crabs want to estimate the percent of crabs in a certain area that are longer than 35 centimeters. The biologists will select a random sample of crabs to measure. Which of the following is the most appropriate method to use for such an estimate?

• A one-sample z-interval for a population proportion

• A one-sample z-interval for a sample proportion

• A two-sample z-interval for a population proportion

• A two-sample z-interval for a difference between population proportions

• A two-sample z-interval for a difference between sample proportions

A random sample of 500 adults living in a large county was selected and 304 adults from the sample indicated that the unemployment rate was of great concern. What is the standard error of the sample proportion $ \hat{p} $?

Which of the following defines a 90 percent confidence interval for the proportion of all students in her district who read at least 1 book last month?

A town council wants to estimate the proportion of residents who are in favor of a proposal to upgrade the computers in the town library. A random sample of 100 residents was selected, and 97 of those selected indicated that they were in favor of the proposal. Is it appropriate to assume that the sampling distribution of the sample proportion is approximately normal?

• No, because the sample is not large enough to satisfy the normality conditions.

• No, because the size of the population is not known.

• Yes, because the sample was selected at random

• Yes, because sampling distributions of proportions are modeled with a normal model

• Yes, because the sample is large enough to satisfy the normality conditions.

A marketing representative wants to estimate the proportion of people in a state who like the new design on the packaging of a certain cleaning product. The representative interviewed 100 people at a certain supermarket, and 82 people indicated that they liked the new design. Have the conditions for creating a confidence interval for the population proportion been met?

• Yes, because the sample was selected at random.

• Yes, because sampling distributions of population proportions are modeled with a normal model.

• Yes, because the sample is large enough to satisfy the normality conditions.

• No, because the sample is not large enough to satisfy the normality conditions.

• No, because the sample may not be representative of all people in the state.

Sue and Javier are working on a statistics project to estimate the proportion of students at their school who have a pet dog. Sue selects a random sample of 81 students from the 2,400 students at their school, and Javier selects a separate random sample of 64 students. They will both construct a 90 percent confidence interval from their estimates. Consider the situation in which the sample proportion from Sue’s sample is equal to the sample proportion from Javier’s sample. Which of the following statements correctly describes their intervals?

• Javier’s interval will have a greater degree of confidence than Sue’s interval will.

• Sue’s interval will have a greater degree of confidence than Javier’s interval will.

• The width of Sue’s interval will be the same as the width of Javier’s interval.

• The width of Sue’s interval will be wider than the width of Javier’s interval.

• The width of Sue’s interval will be narrower than the width of Javier’s interval.

Suppose a 90 percent confidence interval to estimate a population proportion was calculated from a sample proportion of 18 percent and a margin of error of 4 percent. What is the width of the confidence interval?

• 2 percent

• 4 percent

• 8 percent

• 16 percent

• 36 percent

Suppose a researcher wants to use a confidence interval to estimate an unknown population proportion p. Which of the following is not a correct statement?

• The endpoints of the interval can vary with each new sample.

• The probability that p is in the interval is equal to the level of confidence for the interval.

• Whether the interval captures p is not known with certainty.

• The population proportion p is fixed, but the sample proportion p̂ can vary from sample to sample.

• The interval either does or does not capture p.

A random sample of 83 residents of a certain town were asked whether they approve of a proposal to improve the town’s aging bridges. The 95 percent confidence interval to estimate the proportion of all residents of the town who approve of the proposal was calculated to be (0.361, 0.579). Which of the following is a correct interpretation of the interval?

• There is a 0.95 probability that the proportion of all residents in the town who favor the proposal will be between 0.361 and 0.579.

• The probability that 95 percent of the residents in the town will favor the proposal is between 0.361 and 0.579.

• We are 95 percent confident that any sample of 83 residents will produce a sample proportion between 0.361 and 0.579.

• We are 95 percent confident that the proportion of all residents in the sample who favor the proposal is between 0.361 and 0.579.

• We are 95 percent confident that the proportion of all residents in the town who favor the proposal is between 0.361 and 0.579.