Cumulative Exam — Cumulative exam Answers

Twenty-four pairs of adult brothers and sisters were sampled at random from a population. The difference in heights, recorded in inches (brother’s height minus sister’s height), was calculated for each pair. The 95% confidence interval for the mean difference in heights for all brother-and-sister pairs in this population was (–0.76, 4.34). What was the sample mean difference from these 24 pairs of siblings?

A local school board claims that there is a difference in the proportions 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.Based on the 90% confidence interval, (0.03, 0.36), is there convincing evidence of a difference in the true proportions of households, those with school-aged children and those without school-aged children, who would support starting school early?

An animal rescue agent wanted to estimate the true proportion of all animals in shelters that are adopted each month. To do so, she selects a random sample of 100 animals and determines that the 95% confidence interval for the true proportion of animals adopted each month is between 0.12 and 0.24. Which of these statements is a correct interpretation of the confidence level?

A random sample of adults was surveyed about their exercise habits. Of the 100 surveyed, 56 stated they exercise regularly. Which of the following is the 95% confidence interval for p, the proportion of all adults who exercise regularly?Find the z-table here.

The dean of students at a large college is interested in learning about their opinions regarding the percentage of first-year students who should be given parking privileges in the main lot. He sends out an email survey to all students about this issue. A large number of first-year students reply but very few sophomores, juniors, and seniors reply. Based on the responses he receives, he constructs a 90% confidence interval for the true proportion of students who believe first-year students should be given parking privileges in the main lot to be (0.71, 0.79). Which of the following may have an impact on the confidence interval, but is not accounted for by the margin of error?

A teacher has two large containers filled with blue, red, and green beads, and claims the proportions of red beads are the same in each container. Each student shakes the first container, selects 50 beads, counts the number of red beads, and returns the beads to the container. The student repeats this process for the second container. One student’s samples contained 13 red beads from the first container and 16 red beads from the second container.Based on the 95% confidence interval, (–0.24, 0.11), is the teacher’s claim justified?

A local school board wants to determine the proportion of households in the district that would support starting the school year a week earlier. They ask a random sample of 100 households whether they would support starting the school year a week earlier, and 43 households responded that they would. The school board plans to construct a 95% confidence interval for the true proportion of households that would support starting the school year a week earlier. Are the conditions for inference met?

In a small town of 5,832 people, the mayor wants to determine if there is a difference in the proportion of voters ages 18–30 who would support an increase in the food tax, and the proportion of voters ages 31–40 who would support an increase in the food tax. An assistant to the mayor surveys 85 randomly chosen voters ages 18–30, and finds that 62 support the increase. A random sample of 70 voters ages 31–40 is also surveyed, and 56 support the increase. Assuming the conditions for inference have been met, what is the 99% confidence interval for the difference in proportions of voters who would support the increase in the food tax for the different age groups?Find the z-table here.

A statistics class weighed 20 bags of grapes purchased from the store. The bags are advertised to contain 16 ounces, on average. The class calculated the 90% confidence interval for the true mean weight of bags of grapes from this store to be (15.875, 16.595) ounces. Is the store justified in stating that the average weight of the bags of grapes is 16 ounces?

An animal rescue agent wanted to estimate the true proportion of all animals in shelters that are adopted each month. To do so, she selects a random sample of 100 animals that were in shelters last month and determines that the 95% confidence interval for the true proportion of animals adopted is between 0.12 and 0.24. This interval has a margin of error of 0.06. Which of the following can be accounted for by the margin of error?

A school principal claims the graduation rate at a school is 96%. Molly, a student at this school, takes a random sample of students and finds the 95% confidence interval for the true proportion of students graduating from this school is (0.934, 0.983). Is it reasonable to conclude the principal’s claim is incorrect?

A politician estimates that 61% of his constituents will vote for him in the coming election. How many constituents are required for a random sample to obtain a margin of error of at most 0.03 with 95% confidence?Find the z-table here.

A college performs a survey of 424 randomly chosen graduates to estimate the proportion of alumni who are working in the field of their college degree. For example, if a student earned a degree in biology, do they work in the field of biology? Of the 424 alumni, 361 reported that they were working in the field of their college degree. A 98% confidence interval for the true proportion of graduates who are working in the field of their degree is (0.811, 0.892). What is the correct interpretation of the confidence interval?

In a statistics activity, students are asked to determine if there is a difference in the proportion of times that a spinning penny will land with tails up, and the proportion of times a spinning dime will land tails up. The students are instructed to spin the penny and the dime 30 times and record the number of times they land tails up. For one student, the penny lands tails side up 18 times, and the dime lands tails side up 20 times. Assuming the conditions for inference are met, what is the 98% confidence interval for the difference in proportions of tails side up for a penny and a dime?Find the z-table here.

The owner of a popular coffee shop believes that customers who drink coffee are more likely to use their own cup than customers who drink espresso. Customers using their own cups get a 5% discount, which is displayed on the receipt. The owner randomly selects 50 receipts from all coffee purchases and 50 receipts from all espresso purchases. For coffee purchases, 24 receipts showed that the customer used their own cup. For espresso purchases, 18 receipts showed the customer used their own cup.Based on the 99% confidence interval, (–0.13, 0.37), is the coffee shop owner’s claim justified?

In a statistics activity, students are asked to determine if there is a difference in the proportion of times that a spinning penny will land with tails up, and the proportion of times a spinning dime will land tails up. The students are instructed to spin the penny and the dime 30 times and record the number of times they land tails up. For one student, the penny lands tails side up 18 times, and the dime lands tails side up 20 times.Based on the 98% confidence interval, (–0.36, 0.22), is there evidence of a difference in proportions of tails side up for a penny and a dime?

A computer company wants to determine if there is a difference in the proportion of defective computer chips in a day’s production from two different production plants, A and B. A quality control specialist takes a sample of 100 chips from the first hour of production from plant A and determines that there are 12 defective chips. The specialist then takes a sample of 100 chips from the last hour of production from plant B and determines that there are 10 defective chips. He wants to construct a 90% confidence interval for the true difference in proportions of defective chips from a day’s production between the two plants. Are the conditions for inference met?

A newspaper poll found that 54% of the respondents in a random sample of voters in the city plan to vote for candidate Roberts. A 95 percent confidence interval for the population proportion is 0.54 ± 0.06. Based on this interval, what can the newspaper report?

A researcher for a polling organization used a random sample of 1,540 residents in a city to construct a 95 percent confidence interval for the proportion of voters who would vote for candidate Jones. The resulting confidence interval was 0.480 ± 0.025. What is the correct interpretation of the confidence interval?

A quality control inspector selects 12 bottles of apple juice at random from a single day’s production. The mean amount of apple juice in the bottles is 298.3 milliliters, and the 95% confidence interval for the true mean amount of juice dispensed per bottle is (296.4, 300.2). Does this interval give the quality control inspector reason to believe that the mean amount of juice in today’s bottles differs from 300 milliliters, as the juice label promises?