AP统计学讲义课件

,*,*,*,单击此处编辑母版文本样式,第二级,单击此处编辑母版标题样式,统计学精讲精练,主讲老师：黄逸昆,2004,AP STATISTICS FREE-RESPONSE QUESTIONS(FORM B),STATISTICS,SECTION ,Part A,Questions 1-5,Spend about 65 minutes on this part of the exam.,Percent of Section II grade75,1,The Earths Moon has many impact craters that were created when the inner solar system was subjected to heavy bombardment of small celestial bodies. Scientists studied 11 impact craters on the Moon to determine whether there was any relationship between the age of the craters (based on radioactive dating of lunar rocks) and the impact rate (as deduced from the density of the craters). The data are displayed in the scatterplot below.,(,a) Describe the nature of the relationship between impact rate and age.,Prior to fitting a linear regression model, the researchers transformed both impact rate and age by using logarithms. The following computer output and residual plot were produced.,(b) Interpret the value of,r,2,.,(c) Comment on the appropriateness of this linear regression for modeling the relationship between the transformed variables.,2 At a certain university, students who live in the dormitories eat at a common dining hall. Recently, some students have been complaining about the quality of the food served there. The dining hall manager decided to do a survey to estimate the proportion of students living in the dormitories who think that the quality of the food should be improved. One evening, the manager asked the first 100 students entering the dining hall to answer the following question.,(,a) In this setting, explain how bias may have been introduced based on the way this convenience sample was selected and suggest how the sample could have been selected differently to avoid that bias.,(b) In this setting, explain how bias may have been introduced based on the way the question was worded,and,suggest how it could have been worded differently to avoid that bias.,Trains carry bauxite ore from a mine in Canada to an aluminum processing plant in northern New York state in hopper cars. Filling equipment is used to load ore into the hopper cars. When functioning properly, the actual weights of ore loaded into each car by the filling equipment at the mine are approximately normally distributed with a mean of 70 tons and a standard deviation of 0.9 ton. If the mean is greater than 70 tons, the loading mechanism is overfilling.,(a) If the filling equipment is functioning properly, what is the probability that the weight of the ore in a randomly selected car will be 70.7 tons or more? Show your work.,(b) Suppose that the weight of ore in a randomly selected car is 70.7 tons. Would that fact make you suspect that the loading mechanism is overfilling the cars? Justify your answer.,(c) If the filling equipment is functioning properly, what is the probability that a random sample of 10 cars will have a mean ore weight of 70.7 tons or more? Show your work.,(d) Based on your answer in part (c), if a random sample of 10 cars had a mean ore weight of 70.7 tons, would you suspect that the loading mechanism was overfilling the cars? Justify your answer.,4.,The principal at Crest Middle School, which enrolls only sixth-grade students and seventh-grade students, is interested in determining how much time students at that school spend on homework each night. The table below shows the mean and standard deviation of the amount of time spent on homework each night (in minutes) for a random sample of 20 sixth-grade students and a separate random sample of 20 seventh-grade students at this school.,Based on dotplots of these data, it is not unreasonable to assume that the distribution of times for each grade were approximately normally distributed.,(a) Estimate the difference in mean times spent on homework for all sixth- and seventh-grade students in this school using an interval. Be sure to interpret your interval.,(b) An assistant principal reasoned that a much narrower confidence interval could be obtained if the students were paired based on their responses; for example, pairing the sixth-grade student and the seventh-grade student with the highest number of minutes spent on homework, the sixth-grade student and seventh-grade student with the next highest number of minutes spent on homework, and so on. Is the assistant principal correct in thinking that matching students in this way and then computing a matched-pairs confidence interval for the mean difference in time spent on homework is a better procedure than the one used in part (a) ? Explain why or why not.,5. A researcher thinks that modern Thai dogs may be descendants of golden jackals. A random sample of 16 animals was collected from each of the two populations. The length (in millimeters) of the mandible (jawbone) was measured for each animal. The lower quartile, median, and upper quartile for each sample are shown in the table below, along with all values below the lower quartile and all values above the upper quartile.,(a) Display parallel boxplots of mandible lengths (showing outliers, if any) for the modern Thai dogs and the golden jackals on the grid below.,Based on the boxplots, write a few sentences comparing the distributions of mandible lengths for the two types of dogs.,(b) Is it reasonable to use the sample of mandible lengths of modern Thai dogs to construct an interval estimate of the mean mandible length for the population of modern Thai dogs? Justify your answer. (Note: You do not have to compute the interval.),(c) Is it reasonable to use the sample data of mandible lengths of modern Thai dogs and the sample data of mandible lengths of golden jackals to perform a two-sample f-test for the difference in mean mandible lengths for the two types of dogs? Justify your answer. (Note: You do not have to conduct the test.),2004 AP STATISTICS FREE-RESPONSE QUESTIONS(FORM B),STATISTICS,SECTION ,Part B,Questions 6,Spend about 25 minutes on this part of the exam.,Percent of Section II grade25,6. In order to monitor the populations of birds of a particular species on two islands, the following procedure was implemented.,Researchers captured an initial sample of 200 birds of the species on Island A; they attached leg bands to each of the birds, and then released the birds. Similarly, a sample of 250 birds of the same species on Island B was captured, banded, and released. Sufficient time was allowed for the birds to return to their normal routine and location.,Subsequent samples of birds of the species of interest were then taken from each island. The number of birds captured and the number of birds with leg bands were recorded. The results are summarized in the following table.,Assume that both the initial sample and the subsequent samples that were taken on each island can be regarded as random samples from the population of birds of this species.,(a) Do the data from the subsequent samples indicate that there is a difference in proportions of the banded birds on these two islands? Give statistical evidence to support your answer.,(b) Researchers can estimate the total number of birds of this species on an island by using information on the number of birds in the initial sample and the proportion of banded birds in the subsequent sample. Use this information to estimate the total number of birds of this species on Island A. Show your work.,(c) The analyses in parts (a) and (b) assume that the samples of birds captured in both the initial and subsequent samples can be regarded as random samples of the population of birds of this species that live on the respective islands. This is a common assumption made by wildlife researchers. Describe two concerns that should be addressed before making this assumption.,2006 AP STATISQUESTIONS,STATISTICS,SECTION ,Part A,Questions 1-5,Spend about 65 minutes on this part of the exam.,Percent of Section II grade75,TICS FREE-RESPONSE,1. Two parents have each built a toy catapult for use in a game at an elementary school fair. To play the game, students will attempt to launch Ping-Pong balls from the catapults so that the balls land within a 5-centimeter band. A target line will be drawn through the middle of the band, as shown in the figure below. All points on the target line are equidistant from the launching location.,If a ball lands within the shaded band, the student will win a prize.,The parents have constructed the two catapults according to slightly different plans. They want to test these catapults before building additional ones. Under identical conditions, the parents launch 40 Ping-Pong balls from each catapult and measure the distance that the ball travels before landing. Distances to the nearest centimeter are graphed in the dotplots below.,(a) Comment on any similarities and any differences in the two distributions of distances traveled by balls launched from catapult A and catapult B.,(,b) If the parents want to maximize the probability of having the Ping-Pong balls land within the band, which one of the two catapults, A or B, would be better to use than the other? Justify your choice.,(c) Using the catapult that you chose in part (b), how many centimeters from the target line should this catapult be placed? Explain why you chose this distance.,2. A manufacturer of dish detergent believes the height of soapsuds in the dishpan depends on the amount of detergent used. A study of the suds heights for a new dish detergent was conducted. Seven pans of water were prepared. All pans were of the same size and type and contained the same amount of water. The temperature of the water was the same for each pan. An amount of dish detergent was assigned at random to each pan, and that amount of detergent was added to the pan. Then the water in the dishpan was agitated for a set amount of time, and the height of the resulting suds was measured.,A plot of the data and the computer output from fitting a least squares regression line to the data are shown below.,Write the equation of the fitted regression line. Define any variables used in this equation.,(b) Note that,s = 1.99821 in the computer output. Interpret this value in the context of this study.,(c) Identify and interpret the standard error of the slope.,3. The depth from the surface of Earth to a refracting layer beneath the surface can be estimated using methods developed by seismologists. One method is based on the time required for vibrations to travel from a distant explosion to a receiving point. The depth measurement (M) is the sum of the true depth (D) and the random measurement error,(E). That is, M - D + E. The measurement error (E) is assumed to be normally distributed with mean 0 feet and standard deviation 1.5 feet.,(a) If the true depth at a certain point is 2 feet, what is the probability that the depth measurement will be negative?,(b) Suppose three independent depth measurements are taken at the point where the true depth is 2 feet. What is the probability that,at least one of these measurements will be negative?,(c) What is the probability that the,mean of the three independent depth measurements taken at the point where the true depth is 2 feet will be negative?,4. Patients with heart-attack symptoms arrive at an emergency room either by ambulance or self-transportation provided by themselves, family, or friends. When a patient arrives at the emergency room, the time of arrival is recorded. The time when the patients diagnostic treatment begins is also recorded.,An administrator of a large hospital wanted to determine whether the mean wait time (time between arrival and diagnostic treatment) for patients with heart-attack symptoms differs according to the mode of transportation. A random sample of 150 patients with heart-attack symptoms who had reported to the emergency room was selected. For each patient, the mode of transportation and wait time were recorded. Summary statistics for each mode of transportation are shown in the table below.,(a) Use a 99 percent confidence interval to estimate the difference between the mean wait times for ambulance-transported patients and self-transported patients at this emergency room.,(b) Based only on this confidence interval, do you think the difference in the mean wait times is statistically significant? Justify your answer.,5. A biologist is interested in studying the effect of growth-enhancing nutrients and different salinity (salt) levels in water on the growth of shrimps. The biologist has ordered a large shipment of young tiger shrimps from a supply house for use in the study. The experiment is to be conducted in a laboratory where 10 tiger shrimps are placed randomly into each of 12 similar tanks in a controlled environment. The biologist is planning to use 3 different growth-enhancing nutrients (A, B, and C) and two different salinity levels (low and high).,(a) List the treatments that the biologist plans to use in this experiment.,(b) Using the treatments listed in part (a), describe a completely randomized design that will allow the biologist to compare the shrimps growth after 3 weeks.,(c) Give one,statistical,advantage to having only tiger shrimps in the experiment. Explain why this is an advantage.,(d) Give one,statistical,disadvantage to having only tiger shrimps in the experiment. Explain why this is a disadvantage.,2006 AP STATISTICS FREE-RESPONSE QUESTIONS,STATISTICS,SECTION ,Part B,Questions 6,Spend about 25 minutes on this part of the exam.,Percent of Section II grade25,6. A manufacturer of thermostats is concerned that the readings of its thermostats have become less reliable (more variable). In the past, the variance has been 1.52 degrees Fahrenheit (F) squared. A random sample of 10 recently manufactured thermostats was selected and placed in a room that was maintained at 68F. The readings for those 10 thermostats are given in the table below.,(a) State the null and alternative hypotheses that the manufacturer is interested in testing.,(b) Calculate the value of (n-1)S,2,/1.52 for these data.,(c),Assume that the population of thermostat temperatures follows a normal distribution. Use the test statistic,(n-1)S,2,/1.52 from part (b) and the chi-square distribution to test the hypotheses in part (a).,(d) For the test conducted in part (c), what is the smallest value of the test statistic that would have led to the rejection of the null hypothesis at the 5 percent significance level?,Mark this value of the test statistic on the graph of the chi-square distribution below. Indicate the region that contains all of the values that would have led to the rejection of the null hypothesis.,(e) Using simulation, 1,000 samples, each of size 10, were randomly generated from 3 populations with,different variances. Each population was normally distributed with mean 68 and variance greater than 1.52.,The histograms below show the simulated sampling distribution of (n-1)S,2,/1.52 for each population.,Mark the region identified in part (d) on each of the histograms below.,2006 AP STATISTICS FREE-RESPONSE QUESTIONS,(f) Based on the regions that you marked in part (e), identify the simulated sampling distribution that,corresponds to the population with the largest variance. Then identify the simulated sampling distribution that corresponds to the population with the smallest variance. Justify your choices,.,2005 AP STATISTICS FREE-RESPONSE QUESTIONS,STATISTICS,SECTION ,Part A,Questions 1-5,Spend about 65 minutes on this part of the exam.,Percent of Section II grade75,1. The goal of a nutritional study was to compare the caloric intake of adolescents Irving in rural areas of the United States with the caloric intake of adolescents living in urban areas of the United States. A random sample of ninth-grade students from one high school in a rural area was selected. Another random sample of ninth graders from one high school in an urban area was also selected. Each student in each sample kept records of all the food he or she consumed in one day.,The back-to-back stemplot below displays the number of calories of food consumed per kilogram of body weight for each student on that day.,(a) Write a few sentences comparing the distribution of the daily caloric intake of ninth-grade students in the rural high school with the distribution of the daily caloric intake of ninth-grade students in the urban high school.,(b) Is it reasonable to generalize the findings of this study to all rural and urban ninth-grade students in the United States? Explain.,Researchers who want to conduct a similar study are debating which of the following two plans to use.,Plan I: Have each student in the study record all the food he or she consumed in one day. Then,researchers would compute the number of calories of food consumed per kilogram of body weight for each student for that day.,Plan II: Have each student in the study record all the food he or she consumed over the same 7-day period. Then researchers would compute the average daily number of calories of food consumed per kilogram of body weight for each student during that 7-day period.,Assuming that the students keep accurate records, which plan, I or II, would better meet the goal of the study? Justify your answer.,2. Let the random variable,X,represent the number of telephone lines in use by the technical support center of a software manufacturer at noon each day. The probability distribution of,X,is shown in the table below.,(a) Calculate the expected value (the mean) of,X.,(b) Using past records, the staff at the technical support center randomly selected 20 days and found that an average of 1.25 telephone lines were in use at noon on those days. The staff proposes to select another random sample of 1,000 days and compute the average number of telephone lines that were in use at noon on those days. How do you expect the average from this new sample to compare to that of the first sample? Justify your response.,(c) The median of a random variable is defined as any value,x,such that,P(X ,0.5 and,P(X x) ,0.5. For the probability distribution shown in the table above, determine the median of,X.,(d) In a sentence or two, comment on the relationship between the mean and the median relative to the shape of this distribution.,3. The Great Plains Railroad is interested in studying how fuel consumption is related to the number of railcars for its trains on a certain route between Oklahoma City and Omaha.,A random sample of 10 trains on this route has yielded the data in the table below.,A scatterplot, a residual plot, and the output from the regression analysis for these data are shown below.,(a) Is a linear model appropriate for modeling these data? Clearly explain your reasoning.,(b) Suppose the fuel consumption cost is $25 per unit. Give a point estimate (single value) for the change in the average cost of fuel per mile for each additional railcar attached to a train. Show your work.,(c)Interpret the value of,r,2,in the context of this problem.,(d) Would it be reasonable to use the fitted regression equation to predict the fuel consumption for a train on this route if the train had 65 railcars? Explain.,4 Some boxes of a certain brand of breakfast cereal include a voucher for a free video rental inside the box. The company that makes the cereal claims that a voucher can be found in 20 percent of the boxes. However, based on their experiences eating this cereal at home, a group of students believes that the proportion of boxes with vouchers is less than 0.2. This group of students purchased 65 boxes of the cereal to investigate the companys claim. The students found a total of 11 vouchers for free video rentals in the 65 boxes.,Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this ce