AP STAT -- unit 6 MATH/STATISTICS ALL SOLUTION LATEST EDITION 2023/24 GUARANTEED GRADE A+

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? A Yes, all conditions for inference have been met. B No, because the sample size is not large enough to satisfy the conditions for normality. C No, because the sample was not selected at random. D No, because the sample size is not less than 10 percent of the population size. E No, because the sample is not representative of the population. Answer D Correct. The sample size of 30 is greater than 10 percent of the population of 200 and risks violating the independence condition. 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? A No, because the size of the population is not known. B No, because the sample is not large enough to satisfy the normality conditions. C Yes, because the sample is large enough to satisfy the normality conditions. D Yes, because the sample was selected at random. E Yes, because sampling distributions of proportions are modeled with a normal model. Answer B Correct. The number of successes (people who ride the subway) is greater than or equal to 10. However, because the number of failures, i.e.i.e., the number of people who did not ride the subway (2), is less than 10, the conditions for normality are not met. 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 A two-sample zz-interval for a difference between sample proportions B A two-sample zz-interval for a difference between population proportions C A one-sample zz-interval for a sample proportion D A one-sample zz-interval for a population proportion E A one-sample zz-interval for a difference between population proportions Answer D 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 subscribers who approve of the new format. 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. Which of the following is the most appropriate method for creating such an estimate? A A one-sample zz-interval for a sample proportion B A two-sample zz-interval for a difference between population proportions C A two-sample zz-interval for a population proportion D A one-sample zz-interval for a difference between population proportions E A one-sample zz-interval for a population proportion Answer E 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 district residents who favor building a new school. 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 ? A 0.275±1.96(22)(58)80−−−−−−√0.275±1.96(22)(58)80 B 0.22±2.576(0.275)(0.725)80−−−−−−−−−√0.22±2.576(0.275)(0.725)80 C 0.275±2.576(0.275)(0.725)80−−−−−−−−−√0.275±2.576(0.275)(0.725)80 D 0.275±1.96(0.275)(0.725)80−−−−−−−−−√0.275±1.96(0.275)(0.725)80 E 0.22±2.323(0.275)(0.725)80−−−−−−−−−√ Answer C Correct. The sample proportion is 2280=0.=0.275, and the zz-value used to construct a 99 percent confidence interval is 2.576. The confidence interval is pˆ±z∗(pˆ)(1−pˆ)n−−−−−−−√=0.275±2.576(0.275)(0.725)80−−−−−−−−−√p^±z*(p^)(1−p^)n=0.275±2.576(0.275)(0.725)80. 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? A 73 B 97 C 271 D 385 E 1,537 Answer D Correct. To find the least sample size, the formula for margin of error, ME=z∗pˆ(1−pˆ)n−−−−−√ME=z*p^(1−p^)n, can be rearranged to solve for nn. Because a proportion is not known, the value of 0.5 is used for the sample proportion. The zz-value needed for 95% confidence is 1.96. For a 5% margin of error, n=0.25(1.960.05)2=384.16n=0.25(1.960.05)2=384.16. The least value on the list not less than 384.16 is 385. 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 Yes, because the sample was selected at random. B Yes, because sampling distributions of proportions are modeled with the normal model. C Yes, because the sample is large enough to satisfy the normality conditions. D No, because the sample is not large enough to satisfy the normality conditions. E No, because the sample was not selected using a random method. Answer E Correct. The sample should have been chosen randomly from all students at the school, not just from the senior English classes. 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 (0.313,0.447)(0.313,0.447) B (0.324,0.436)(0.324,0.436) C (65,87)(65,87) D (1565,2235)(1565,2235) E (1620,2180) Answer E Correct. 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. 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? A A two-sample zz-interval for a population proportion B A one-sample zz-interval for a sample proportion C A one-sample zz-interval for a population proportion D A two-sample zz-interval for a difference between sample proportions E A two-sample zz-interval for a difference between population proportions Answer C 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. 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? A We are 95% confident that the proportion who intend to vote for Candidate R from the random sample is between 42% and 52%. B We are 95% confident that the proportion who intend to vote for Candidate R from the population is between 42% and 52%. C We are 95% confident that the proportion who intend to vote for Candidate R from the random sample is 47%. D We are 95% confident that the proportion who intend to vote for Candidate R from the population is 47%. E We are confident that 95% of the population intend to vote for Candidate R. Answer B Correct. 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. 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? A More than 90 percent of the residents support the increase. B More than 60 percent of the residents support the increase. C More than 40 percent of the residents support the increase. D Fewer than 10 percent of the residents support the increase. E Fewer than 25 percent of the residents support the increase. Answer C Correct. 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. 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? A We are 95 percent confident that any sample of 1,175 people will produce a sample proportion between 0.146 and 0.214. B 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. C We are 95 percent confident that the proportion of people in the sample who think climate change is a problem is between 0.146 and 0.214. D The probability that 95 percent of all people in the country who think climate change is a problem is between 0.146 and 0.214. E The probability is 0.95 that the proportion of all people in the country who think climate change is a problem is between 0.146 and 0.214. Answer B Correct. 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. 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? A The midpoint of Elly's interval will be greater than the midpoint of Drew's interval. B The midpoint of Elly's interval will be less than the midpoint of Drew's interval. C The width of Elly's interval will be greater than the width of Drew's interval. D The width of Elly's interval will be less than the width of Drew's interval. E The width of Elly's interval will be equal to the width of Drew's interval. Answer D Correct. 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. 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? A 2.5 percent B 5 percent C 10 percent D 20 percent E 45 percent Answer B Correct. The margin of error is one-half of the total width of the confidence interval, and one-half of 10 percent is 5 percent. 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? A Less than half of all women do not find time to focus on their own health. B More than 40 percent of all women do not find time to focus on their own health. C Approximately 45 percent of all women do not find time to focus on their own health. D More than 45 percent of all women do not find time to focus on their own health. E More than 25 percent of all women do not find time to focus on their own health. Answer D Correct. 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. 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? A There is approximately a 90 percent chance that pp is contained in the interval. B There is approximately a 90 percent chance that a randomly selected proportion pˆp^ will be contained in the interval. C Approximately 90 percent of all possible sample proportions pˆp^ will be contained in the interval. D In repeated samplings with the same sample size, approximately 90 percent of the intervals created will capture the population proportion pp. E In repeated samplings with the same sample size, approximately 90 percent of the intervals created will capture the sample proportion pˆp^. Answer D Correct. The confidence level of 90 percent reflects the percent of all possible intervals that will capture the population parameter pp. 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. Based on the confidence interval, which of the following claims is supported? A Less than 25 percent of all adults find the word “whatever” to be the most annoying word in conversation. B More than 30 percent of all adults find the word “whatever” to be the most annoying word in conversation. C More than 45 percent of all adults find the word “whatever” to be the most annoying word in conversation. D More than half of all adults find the word “whatever” to be the most annoying word in conversation. E At least 95 percent of all adults find the word “whatever” to be the most annoying word in conversation. Answer B Correct. 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. 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? A Lila’s sample size is most likely greater than Robert’s sample size. B Robert’s sample size is mostly likely greater than Lila’s sample size. C Lila and Robert will both find the same sample proportion of students who have seen the movie. D Lila’s interval has a greater degree of confidence than that of Robert. E Robert’s interval has a greater degree of confidence than that of Lila. Answer A Correct. 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. In order to make statistical inferences when testing a population proportion p, which of the following conditions verify that inference procedures are appropriate? The data are collected using a random sample or random assignment. The sample size is less than 10 percent of the population size. np0≥10 and n(1−p0)≥10 for sample size n and hypothesized proportion p0. A II only B IIII only C IIIIII only D IIII and IIIIII only E II, IIII, and III Answer E Correct. All three conditions (randomization, independence, and normality) need to be met to make statistical inferences when testing a population proportion. In a population of bats living in a certain region, 30 percent have a wingspan greater than 10 inches. In a random sample of 80 bats living outside of the region, 20 had a wingspan greater than 10 inches. Consider a one-sample z-test to investigate whether there is evidence that the proportion of bats with a wingspan greater than 10 inches living outside the region is different from that of the bats living in the region. Which of the following is the correct test statistic? A z=0.30−0.25(0.25)(0.75)80√z=0.30−0.25(0.25)(0.75)80 B z=0.30−0.25(0.30)(0.70)80√z=0.30−0.25(0.30)(0.70)80 C z=0.20−0.30(0.30)(0.70)80√z=0.20−0.30(0.30)(0.70)80 D z=0.25−0.30(0.25)(0.75)80√z=0.25−0.30(0.25)(0.75)80 E z=0.25−0.30(0.30)(0.70)80√ Answer E Correct. The sample proportion is 2080=0.=0.25, and the test statistic is given by z=pˆ−p0p0(1−p0)n√=0.25−0.30(0.30)(0.70)80√z=p^−p0p0(1−p0)n=0.25−0.30(0.30)(0.70)80. The germination rate is the rate at which plants begin to grow after the seed is planted. A seed company claims that the germination rate for their seeds is 90 percent. Concerned that the germination rate is actually less than 90 percent, a botanist obtained a random sample of seeds, of which only 80 percent germinated. What are the correct hypotheses for a one-sample z-test for a population proportion p ? A H0:p=0.80Ha:p0.80H0:p=0.80Ha:p0.80 B H0:p=0.80Ha:p0.80H0:p=0.80Ha:p0.80 C H0:p=0.90Ha:p0.90H0:p=0.90Ha:p0.90 D H0:p=0.90Ha:p0.90H0:p=0.90Ha:p0.90 E H0:p=0.90Ha:p≠0.90 Answer C Correct. The null hypothesis is a statement about the population proportion, which in this case is 0.90. The alternative hypothesis is the botanist's concern that the population proportion is less than 0.90. A one-sample z-test for a population proportion will be conducted using a simple random sample selected without replacement from a population. Which of the following is a check for independence? A np0≥10np0≥10 and n(1−p0)≥10n(1−p0)≥10 for sample size nn and population proportion p0p0. B Each sample proportion value is less than or equal to 0.5. C The sample size is more than 10 times the population size. D The population size is more than 10 times the sample size. E The population distribution is approximately normal. Answer D Correct. When sampling without replacement, the lack of independence can be ignored if the sample size is small relative to the population size (less than 10%)-that is, if the population size is more than 10 times the sample size. After a tropical storm in a certain state, news reports indicated that 19 percent of households in the state lost power during the storm. A state engineer believes that estimate is too low. The engineer will collect data to perform a hypothesis test on the proportion of all households without power. Which of the following are the appropriate hypotheses for such a test? A H0:pˆ=0.19Ha:pˆ0.19H0:p^=0.19Ha:p^0.19 B H0:pˆ=0.19Ha:pˆ0.19H0:p^=0.19Ha:p^0.19 C H0:p=0.19Ha:p0.19H0:p=0.19Ha:p0.19 D H0:p=0.19Ha:p≠0.19H0:p=0.19Ha:p≠0.19 E H0:p=0.19Ha:p0.19H0:p=0.19Ha:p0.19 Answer E Correct. The engineer believes the actual proportion is higher, so the alternative hypothesis is greater than the population proportion. Approximately 38 percent of people living in Region W have the blood type O positive. A random sample of 100 people from Region X revealed that 35 people in the sample had the blood type O positive. Consider a hypothesis test to investigate whether the percent of people in Region X with O positive blood is different from that of in Region W. Which of the following is the appropriate null hypothesis for the investigation? A H0:pˆ=0.35H0:p^=0.35 B H0:pˆ=0.38H0:p^=0.38 C H0:p=0.35H0:p=0.35 D H0:p=0.38H0:p=0.38 E H0:p≠0.38 Answer D Correct. The null hypothesis is a statement of the population proportion, which in this case is 0.38. Consider a population with population proportion p, and a sample from the population with sample proportion pˆ. Which of the following describes the purpose of the one-sample z-test? A To estimate the value of pˆp^ B To estimate the value of pp C To estimate a margin of error for pˆp^ D To estimate the probability of observing a value as extreme as pˆp^ given pp E To estimate the probability of observing a value as extreme as pp given pˆ Answer D Correct. The test statistic for a one-sample zz-test is the distance, in units of standard deviations, between the statistic and the given parameter. From that distance, probabilities (a p-value) can be calculated and a claim can be assessed. We have an expert-written solution to this problem! Studies indicate that about 10 percent of polar bears weigh more than 1,000 pounds. A biologist studying the bears thinks that percent might be too high. From a random sample of polar bears, the biologist found only 8 percent of the sample weighing over 1,000 pounds. Which of the following is the most appropriate method for the biologist’s study? A A one-sample zz-test for a sample proportion B A one-sample zz-test for a population proportion C A one-sample zz-test for a difference in population proportions D A two-sample zz-test for a difference in sample proportions E A two-sample zz-test for a difference in population proportions Answer B Correct. A one-sample zz-test for a population proportion is appropriate to investigate how likely 0.08 is just by chance, assuming the population proportion is 0.10. In high school X, approximately 9 percent of the students saw a certain movie on opening night. From a random sample of 200 students from high school Y, 22 saw the movie on opening night. Consider a hypothesis test to investigate whether the proportion of all students in high school Y who saw the movie on opening night is greater than that of high school X. Which of the following is the standard deviation used to calculate the test statistic for the one-sample z-test? A (0.11)(0.89)200−−−−−−−−√(0.11)(0.89)200 B (0.09)(0.91)200−−−−−−−−√(0.09)(0.91)200 C (22)(178)200−−−−−−√(22)(178)200 D (0.11)(0.89)200√(0.11)(0.89)200 E (0.09)(0.91)200√ Answer B Correct. The standard deviation uses the value of the population proportion 0.09 and is given by σpˆ=p0(1−p0)n−−−−−−√=(0.09)(0.91)200−−−−−−−−√σp^=p0(1−p0)n=(0.09)(0.91)200. A recent study indicated that 17 percent of adults in the country actively seek out science news sites to keep current on topics in science. A university researcher believes that percent is too low. From a random sample of adults in the country, the researcher found that 22 percent of the sample actively seek out science news sites. Which of the following is the most appropriate method for the researcher’s study? A A two-sample zz-test for a difference in population proportions B A two-sample zz-test for a difference in sample proportions C A one-sample zz-test for a difference in population proportions D A one-sample zz-test for a sample proportion E A one-sample zz-test for a population proportion Answer E Correct. A one-sample zz-test for a population proportion is appropriate to investigate whether the sample proportion 0.22 provides sufficient evidence that the population proportion is greater than 0.17. A significance test is conducted for which the alternative hypothesis states that more than 85 percent of adult sea turtles on a certain beach are female. The p-value for the test is 0.4158. If the null hypothesis is true, which of the following statements is a correct interpretation of the p-value? A Of all possible samples of the same size, 41.58 percent will result in 85 percent of adult sea turtles on the beach being female. B Of all possible samples of the same size, 41.58 percent will result in 85 percent or more of adult sea turtles on the beach being female. C Of all possible samples of the same size, 41.58 percent will result in 85 percent or less of adult sea turtles on the beach being female. D Of all possible samples of the same size, 20.79 percent will result in 85 percent or more of adult sea turtles on the beach being female. E Of all possible samples of the same size, 20.79 percent will result in 85 percent of adult sea turtles on the beach being female. Answer B Correct. The pp-value represents the probability of obtaining a proportion as extreme as or more extreme than the sample proportion. Past studies indicate that about 60 percent of the trees in a forested region are classified as softwood. A botanist studying the region suspects that the proportion might be greater than 0.60. The botanist obtained a random sample of trees from the region and conducted a test of H0:p=0.6 versus Ha:p0.6. The p-value of the test was 0.015. Which of the following is a correct interpretation of the p-value? A If it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is the probability of obtaining a population proportion greater than 0.6. B If it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is the probability of obtaining a sample proportion as small as or smaller than the one obtained by the botanist. CONTINUED..