Estimating the Difference between Two Population Proportions — Unit test Answers

A teacher has two large containers filled with blue, red, and green beads. He wants his students to estimate the difference in the proportion of red beads in each container. Each student shakes the first container, selects 25 beads, counts the number of red beads, and returns the beads to the container. The students repeat this process for the second container. One student sampled 10 red beads from the first container and 8 red beads from the second container. The students are asked to construct a 95% confidence interval for the difference in proportions of red beads in each container. Are the conditions for inference met?

A politician claims that a proposal for a new traffic law is broadly supported by both political parties and that a person from either political party is equally likely to support the proposed legislation. He cites two recent polls that said 70% of a random sample of 550 people from his political party supports the law, and 65% of a random sample of 420 people from the other political party supports the law. The 95 percent confidence interval for the difference in population proportions is (–0.010, 0.110). Based on the interval, is the politician’s claim justified?

A political candidate feels that she performed particularly well in the most recent debate against her opponent. Her campaign manager polled a random sample of 400 likely voters before the debate and a random sample of 500 likely voters after the debate. The 95% confidence interval for the true difference (post-debate minus pre-debate) in proportions of likely voters who would vote for this candidate was (–0.014, 0.064). What is the correct interpretation of the 95 percent confidence interval?

The nutrition supervisor for a school district is considering adding a baked potato bar to the lunch for all the high school cafeterias. He wants to determine if there is a difference in the proportion of students who would purchase from the potato bar for two high schools, East and West. The cafeteria manager at each high school randomly surveys 90 students. At East High School, 63 of the students said they would purchase the potato bar on that day. At West High School, 45 students said they would.Based on the 99% confidence interval, (0.02, 0.38), is there convincing evidence of a difference in the proportions of students who would purchase from the potato bar between East and West High Schools?

A local dentist is concerned that less than half of her patients floss daily. A 95% confidence interval for the true proportion of her patients who floss daily is (0.325, 0.701). Is it reasonable to believe that less than half of her patients floss daily?

What is the z* critical value for constructing a 92% confidence interval for a proportion?Find the z-table here.

A student believes that a certain number cube is unfair and is more likely to land with a six facing up. The student rolls the number cube 15 times, and finds that the cube lands with a six facing up five times. The student wants to construct a 99% confidence interval for the proportion of times this number cube lands with a six facing up. Are the conditions for inference met?

A teacher would like to estimate the mean number of steps students take during the school day. To do so, she selects a random sample of 50 students and gives each one a pedometer at the beginning of the school day. They wear the pedometers all day and then return them to her at the end of the school day. From this, she computes the 98% confidence interval for the true mean number of steps students take during the school day to be 8,500 to 10,200. If the teacher had used a 90% confidence interval rather than a 98% confidence interval, what would happen to the width of the interval?

A statistics student from a large high school takes a random sample of 200 students and finds that 123 are actively involved in a political party. A 99% confidence interval for the proportion of students at this school who are actively involved in a political party is (0.526, 0.704). Which statement correctly interprets the interval?

The maker of a cell phone screen protector would like to estimate the proportion of customers who file a warranty claim. To do so, they select a random sample of 200 customers and determine that the 96% confidence interval for the true proportion of customers who file a warranty claim to be 0.15 to 0.28. Which of the following can be accounted for by the margin of error of this interval?

A student wants to survey the sophomore class of 200 students about whether the school should require uniforms. A random sample of 50 sophomores is surveyed and asked whether they support the school adopting uniforms. Of the 50 sophomores, 12 say they would favor school uniforms. Assuming the conditions for inference have been met, what is the 90% confidence interval for the true proportion of sophomores who favor the adoption of uniforms?

A teacher has a large container filled with blue, red, and green beads. She wants her students to estimate the proportion of red beads. Each student shakes the container, selects 50 beads, counts the number of red beads, and returns the beads to the container. One student’s sample contained 19 red beads. The students are asked to construct a 95% confidence interval for the true proportion of red beads in the container. Are the conditions for inference met?

A computer company wants to determine if there is difference in the proportions of defective computer chips in a day’s production from two different production plants, A and B. A quality control specialist takes a random sample of 100 chips from the day’s production from plant A, and determines that there are 12 defective chips. The specialist then takes a random sample of 100 chips from the day’s production from plant B, and determines that there are 10 defective chips.Based on the 90% confidence interval, (–0.05, 0.09), is there convincing evidence of a difference in the true proportions of defective chips from a day’s production between the two plants?

A large home improvement store is considering expanding its selection of moving products, such as cardboard boxes and packing tape. The store constructs a 90% confidence interval to estimate the proportion of all customers who have moved houses at least once in the last five years. In the random sample of 620 customers, 201 (32.4%) replied that they had moved at least once in the last five years. The sample yielded the 90% confidence interval (0.293, 0.355) for the proportion of all customers who have moved in the past five years. Sheldon, an employee who wishes to expand the selection of moving products, claims that more than 1 in 4 customers have moved in the past five years. Zachary, another employee, urges Sheldon to be bolder in his claim and say that more than 1 in 3 customers have moved in the past five years. Based on the confidence interval, which employee’s claim is plausible?

A political candidate feels that she performed particularly well in the most recent debate against her opponent. Her campaign manager polled a random sample of 400 likely voters before the debate and a random sample of 500 likely voters after the debate. The 95% confidence interval for the true difference (post-debate minus pre-debate) in proportions of likely voters who would vote for this candidate was (–0.014, 0.064). What is the margin of error for this confidence interval?

A principal of a large high school wants to estimate the true proportion of high school students who use the community’s public library. To do so, he selects a random sample of 50 students and asks them if they use the community’s public library. The 95% confidence interval for the true proportion of all students who use the community’s public library is 0.25 to 0.34. If the principal had used a sample size of 200 students rather than 50 students, how would the length of the second interval compare to the original interval?

A principal of a large high school wants to estimate the true proportion of high school students who use the community’s public library. To do so, he selects a random sample of 50 students and asks them if they use the community’s public library. The 95% confidence interval for the true proportion of all students who use the community’s public library is 0.25 to 0.34. Which of these statements is a correct interpretation of the confidence level?

A local school board wants to estimate the difference in the proportion of households with school-aged children that would support starting the school year a week earlier, and the proportion of households without school-aged children that would support starting the school year a week earlier. They survey a random sample of 40 households with school-aged children about whether they would support starting the school year a week earlier, and 30 households respond yes. They survey a random sample of 45 households that do not have school-aged children, and 25 respond yes. Assuming the conditions for inference have been met, what is the 90% confidence interval for the difference in proportions of households that would support starting the school year a week earlier?Find the z-table here.

Natalia estimates that she wins a board game 72% of the time. How large of a random sample is required to obtain a margin of error of at most 0.06 with 95% confidence?Find the z-table here.

A random sample of adults was surveyed about whether they shop online. Of the 90 adults surveyed, 72 stated they shop online. What is the 99% confidence interval for p, the proportion of adults who shop online?Find the z-table here.