Unit 6a - Confidence Intervals for proportions

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?

• one-sample z-interval for a sample proportion

• one-sample z-interval for a population proportion

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

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

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

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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?

• one-sample z-interval for a sample proportion

• one-sample z-interval for a population proportion

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

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

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

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Alma is estimating the proportion of students in her school district who, in the past month, read at least 1 book. From a random sample of 50 students, she found that 32 students read at least 1 book last month. Assuming all conditions for inference are met, 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?

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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.

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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.

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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

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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 (constant), but the sample proportion p-hat can vary from sample to sample.

• The interval either does or does not capture p.

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• 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.

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Surveys were sent to a random sample of owners of all-wheel-drive (AWD) vehicles and to a random sample of owners of front-wheel-drive (FWD) vehicles. The proportion of owners who were satisfied with their vehicles was recorded for each sample. The sample proportions were used to construct the 95 percent confidence interval for a difference in population proportions (FWD minus AWD) for satisfied owners. The interval is given as (−0.01,0.12).

A car company believes that the proportion of satisfied owners of AWD vehicles differs from the proportion of satisfied owners of FWD vehicles. Does the confidence interval provide evidence that this belief is plausible?

• No. The interval contains 0.

• No. Most of the values in the interval are not close to 0.

• No. The value of 0 is not in the middle of the interval.

• Yes. The interval does contain 0.

• Yes. There are more positive values in the interval than negative values.

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