AP统计学讲义（课堂PPT）

1、1统计学精讲精练主讲老师：黄逸昆22004 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 grade7531 The Earths Moon has many impact craters that were created when the inner solar system was subjected to heavy bom

2、bardment 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

3、 the scatterplot below.4(a) Describe the nature of the relationship between impact rate and age.5Prior 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.6(b) Interpret the valu

4、e of r 2.(c) Comment on the appropriateness of this linear regression for modeling the relationship between the transformed variables.72 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 f

5、ood 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

6、.8(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 w

7、orded and suggest how it could have been worded differently to avoid that bias. 9Trains 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

8、 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.10(a) If the filling equipment is functioning properl

9、y, 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 ans

10、wer.(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

11、 that the loading mechanism was overfilling the cars? Justify your answer.114. 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 sh

12、ows 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.12Based on dotplots of these data, it is not unreasonable to assume that the dis

13、tribution 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.13(b) An assistant principal reasoned that a much narro

14、wer 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 num

15、ber 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

16、 why not.145. 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 quarti

17、le for each sample are shown in the table below, along with all values below the lower quartile and all values above the upper quartile.15(a) Display parallel boxplots of mandible lengths (showing outliers, if any) for the modern Thai dogs and the golden jackals on the grid below.16Based on the boxp

18、lots, 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 answ

19、er. (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? Just

20、ify your answer. (Note: You do not have to conduct the test.)172004 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 grade25186. In order to monitor the populations of birds of a particular speci

21、es 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,

22、 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 summariz

23、ed in the following table.19Assume 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 t

24、he banded birds on these two islands? Give statistical evidence to support your answer.20(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.

25、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 speci

26、es 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.21 2006 AP STATISQUESTIONS2006 AP STATISQUESTIONSSTATISTICS STATISTICS SECTION SECTION Part A Part A Questions 1-5 Questions

27、1-5 Spend about 65 minutes on this part of Spend about 65 minutes on this part of the exam. the exam. Percent of Section II grade75Percent of Section II grade75TICS FREE-RESPONSETICS FREE-RESPONSE221. Two parents have each built a toy catapult for use in a game at an elementary school fair. To play

28、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.23If a b

29、all 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

30、 and measure the distance that the ball travels before landing. Distances to the nearest centimeter are graphed in the dotplots below.24(a) Comment on any similarities and any differences in the two distributions of distances traveled by balls launched from catapult A and catapult B.25(b) If the par

31、ents 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 catapu

32、lt be placed? Explain why you chose this distance.262. 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

33、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, a

34、nd the height of the resulting suds was measured.27A plot of the data and the computer output from fitting a least squares regression line to the data are shown below.28Write the equation of the fitted regression line. Define any variables used in this equation.(b) Note that s = 1.99821 in the compu

35、ter output. Interpret this value in the context of this study.(c) Identify and interpret the standard error of the slope.293. 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 requi

36、red 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

37、 feet.30(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

38、 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?314. Patients with heart-attack symptoms arrive at an emergency room either by ambulance or self-transportation provided by themselves

39、, 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.32An administrator of a large hospital wanted to determine whether the mean wait time (time between arrival and diagnostic trea

40、tment) 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

41、 for each mode of transportation are shown in the table below.33(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 t

42、hink the difference in the mean wait times is statistically significant? Justify your answer.345. 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

43、 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 t

44、wo different salinity levels (low and high).(a) List the treatments that the biologist plans to use in this experiment.35(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 statist

45、ical 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.36 2006 AP STATISTICS FREE-2006 AP STATISTICS FREE-RESPONSE QUESTIONSRESPONSE QUEST

46、IONSSTATISTICS STATISTICS SECTION SECTION Part B Part B Questions 6 Questions 6 Spend about 25 minutes on this part of Spend about 25 minutes on this part of the exam. the exam. Percent of Section II grade25Percent of Section II grade25376. A manufacturer of thermostats is concerned that the reading

47、s 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 give

48、n in the table below.38(a) State the null and alternative hypotheses that the manufacturer is interested in testing.(b) Calculate the value of (n-1)S2/1.52 for these data. 39(c) Assume that the population of thermostat temperatures follows a normal distribution. Use the test statistic (n-1)S2/1.52 f

49、rom 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?40Mark this value of the test stati

50、stic 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.41(e) Using simulation, 1,000 samples, each of size 10, were randomly generated from 3 populations withdifferent variances. Each pop

51、ulation was normally distributed with mean 68 and variance greater than 1.52.The histograms below show the simulated sampling distribution of (n-1)S2/1.52 for each population.Mark the region identified in part (d) on each of the histograms below.42 2006 AP STATISTICS FREE-2006 AP STATISTICS FREE-RES

52、PONSE QUESTIONSRESPONSE QUESTIONS43(f) Based on the regions that you marked in part (e), identify the simulated sampling distribution thatcorresponds to the population with the largest variance. Then identify the simulated sampling distribution that corresponds to the population with the smallest va

53、riance. Justify your choices.44 2005 AP STATISTICS FREE-2005 AP STATISTICS FREE-RESPONSE QUESTIONSRESPONSE QUESTIONSSTATISTICS STATISTICS SECTION SECTION Part A Part A Questions 1-5 Questions 1-5 Spend about 65 minutes on this part of Spend about 65 minutes on this part of the exam. the exam. Percen

54、t of Section II grade75Percent of Section II grade7545 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 stud

55、ents 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.46The back-to-back stemplot below displays the number of calo

56、ries of food consumed per kilogram of body weight for each student on that day. 47(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 h

57、igh 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.48Researchers 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

58、 the food he or she consumed in one day. Thenresearchers 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 c

59、ompute 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.49 2. Let the random variable X repr

60、esent 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.50(b) Using past records, the staff at the technical support center r