AP Stats 6.1-6.3 Confidence Intervals

0.275 ±2.576√(0.275)(0.725)/80

6.2 A random sample of 80 people was selected, and 22 of the selected people indicated that it would be a good idea to eliminate the penny from circulation. What is the 99 percent confidence interval constructed from the sample proportion pˆ ?

No, because the sample was not selected using a random method.

6.2 A school librarian wanted to estimate the proportion of students in the school who had read a certain book. The librarian sampled 50 students from the senior English classes, and 35 of the students in the sample had read the book. Have the conditions for creating a confidence interval for the population proportion been met?

A one-sample zz-interval for a population proportion Correct. A zz-interval is used to estimate a population proportion for a categorical variable. In this case, the population proportion is the proportion of all trees in the forest that are infested with the beetle.

6.2 Environmentalists want to estimate the percent of trees in a large forest that are infested with a certain beetle. The environmentalists will select a random sample of trees to inspect. Which of the following is the most appropriate method for creating such an estimate?

385

6.2 Paul will select a random sample of students to create a 95 percent confidence interval to estimate the proportion of students at his college who have a tattoo. Of the following, which is the smallest sample size that will result in a margin of error of no more than 5 percentage points?

(1620,2180) The 90 percent confidence interval for the proportion of people who would indicate they were experiencing side effects from the drug is (0.324,0.436)(0.324,0.436). The interval estimate for the number of people who would indicate they were experiencing side effects from the drug is found by multiplying the endpoints of the interval for the proportions by 5,000.

6.2 Researchers investigating a new drug selected a random sample of 200 people who are taking the drug. Of those selected, 76 indicated they were experiencing side effects from the drug. If 5,000 people took the drug, which of the following is closest to the interval estimate of the number of people who would indicate they were experiencing side effects from the drug at a 90 percent level of confidence?

A one-sample zz-interval for a population proportion

6.2 The manager of a magazine wants to estimate the percent of magazine subscribers who approve of a new cover format. To gather data, the manager will select a random sample of subscribers. Which of the following is the most appropriate interval for the manager to use for such an estimate?

A one-sample zz-interval for a population proportion

6.2 The superintendent of a large school district wants to estimate the percent of district residents who support the building of a new middle school. To gather data, the superintendent will select a random sample of district residents.

We are 95 percent confident that the proportion of all people in the country who think climate change is a problem is between 0.146 and 0.214. The interval is a statement about how confident we are that the interval has captured the population parameter—the proportion of all people in the country who think climate change is a problem.

6.3 A random sample of 1,175 people in a certain country were asked whether they thought climate change was a problem. The sample proportion of those who think climate change is a problem was calculated, and a 95 percent confidence interval was constructed as (0.146,0.214). Which of the following is a correct interpretation of the interval?

More than 30 percent of all adults find the word "whatever" to be the most annoying word in conversation. A claim that the actual percent is greater than 30 percent is supported by the confidence interval. The interval represents plausible values for the population proportion and all values contained in the interval are greater than 0.3.

6.3 A recent study on the way that people talk indicated, with 95 percent confidence, that between 35 percent and 41 percent of all adults find the word "whatever" to be the most annoying word in conversation.

More than 45 percent of all women do not find time to focus on their own health. The claim is not supported by the confidence interval. The percentages contained in the interval are from 42 percent to 48 percent. Percentages less than 45 percent are also plausible values for the population parameter.

6.3 Based on findings from a recent study on women's health, researchers created a 90 percent confidence interval of (0.42,0.48) to estimate the percent of all women who do not find time to focus on their own health. Based on the confidence interval, which of the following claims is not supported?

In repeated samplings with the same sample size, approximately 90 percent of the intervals created will capture the population proportion pp. The confidence level of 90 percent reflects the percent of all possible intervals that will capture the population parameter pp.

6.3 Consider a 90 percent confidence interval for a population proportion p. Which of the following is a correct interpretation of the confidence level 90 percent?

5 percent The margin of error is one-half of the total width of the confidence interval, and one-half of 10 percent is 5 percent.

6.3 Consider a 90 percent confidence interval to estimate a population proportion that is constructed from a sample proportion of 66 percent. If the width of the interval is 10 percent, what is the margin of error?

The width of Elly's interval will be less than the width of Drew's interval. For the same sample, as the confidence level increases, the width of the interval increases. Elly's confidence level (90%) is less than Drew's (99%), so the width of her interval will be less than Drew's.

6.3 Elly and Drew work together to collect data to estimate the percentage of their classmates who own a particular brand of shoe. Using the same data, Elly will construct a 90 percent confidence interval and Drew will construct a 99 percent confidence interval. Which of the following statements is true?

We are 95% confident that the proportion who intend to vote for Candidate R from the population is between 42% and 52% Adding and subtracting the margin of error (5%) from the point estimate (47%) gives an interval from 42% to 52%. The interval is a statement about how confident we are that the interval has captured the population parameter.

6.3 From a random sample of potential voters in an upcoming election, 47% indicated they intended to vote for Candidate R. A 95 percent confidence interval was constructed from the sample, and the margin of error for the estimate was 5%. Which of the following is the best interpretation of the interval?

Lila's sample size is most likely greater than Robert's sample size. Both Lila and Robert use the same level of confidence, but Lila's interval is narrower with a width of 5 percent (0.35−0.30=0.05)(0.35−0.30=0.05) as opposed to Robert's interval width of 7 percent (0.34−0.27=0.07)(0.34−0.27=0.07). Lila's pˆp^ value is 0.325 (the midpoint of her interval), and Robert's pˆp^ value is 0.305 (the midpoint of his interval), so the value of pˆ(1−pˆ)p^(1−p^) in the calculation for the margin of error is very close for Lila's interval and Robert's interval. Therefore, the difference in the confidence interval width is most likely due to Lila's sample size being greater.

6.3 Lila and Robert attend different high schools. They will estimate the population percentage of students at their respective schools who have seen a certain movie. Lila and Robert each select a random sample of students from their respective schools and use the data to create a 95 percent confidence interval. Lila's interval is (0.30,0.35), and Robert's interval is (0.27,0.34). Which of the following statements can be concluded from the intervals?

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

6.2 A city planner wants to estimate the proportion of city residents who commute to work by subway each day. A random sample of 30 city residents was selected, and 28 of those selected indicated that they rode the subway to work. Is it appropriate to assume that the sampling distribution of the sample proportion is approximately normal?

No, because the sample size is not less than 10 percent of the population size.

6.2 A local arts council has 200 members. The council president wanted to estimate the percent of its members who have had experience in writing grants. The president randomly selected 30 members and surveyed the selected members on their grant-writing experience. Of the 30 selected members, 12 indicated that they did have the experience. Have the conditions for inference with a one-sample z-interval been met?

More than 40 percent of the residents support the increase. The claim is supported by the confidence interval. The interval represents plausible values for the population proportion of residents who support the increase and all values in the confidence interval are over 40 percent.

6.3 A random sample of residents in city J were surveyed about whether they supported raising taxes to increase bus service for the city. From the results, a 95 percent confidence interval was constructed to estimate the proportion of people in the city who support the increase. The interval was (0.46,0.52). Based on the confidence interval, which of the following claims is supported?