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AP Statistics Fall Semester Cumulative ReviewThis review is comprised of Chapters 2-6 previous reviewsNote- There was no review for chapter 1: Make sure to review topics involving mean, median, standard deviation, variance, 5 number summary, creating a box plot, the facts behind box plots, stem and leaf plots, histograms, bar graphs, pie charts, calculator commands, CUSS, and two way tables. In exercise 1 and 2 refer to the following setting. According to the National Center for Health Statistics, the distribution of heights for 15 year old males is symmetric, single-peaked, and bell-shaped. For this distribution, a z-score of 0 corresponds to a height of 170 centimeters (cm) and a z-score of 1 corresponds to a height of 177.5 cm. Male Heights consider the height distribution for 15 year old males.Find its mean and standard deviation. Show your method clearlyWhat Height would correspond to a z-score of 2.5? Show your work.Is Paul tall? Paul is 15 years old and 179 cm tall. Find the z-score corresponding to Paul’s height. Explain what this value means.Paul’s height puts him at the 85th percentile among 15 year old males. Explain what this means to someone who knows no puter Use Mrs. Causey asked her students how much time they had spent using a computer during the previous week. The following figure shows a cumulative relative frequency graph of her students’ responses.At what percent does a student who used her computer for 10 hours last week fall?Estimate the median, Q1, and Q3 from the graph. Do you suspect there any outliers, explain?What the mean means The figure below is a density curve. Trace the curve onto our paper.Mark the appropriate location of the median. Justify your choice in locationMark the appropriate location of the mean. Justify your choice in location21240755715000Horse pregnancies Bigger animals tend to carry their young longer before birth. The length of horse pregnancies from conception to birth varies according to a roughly Normal distribution with mean 336 days and standard deviation of 3 days. Use the 68-95-99.7 rule to answer the following questions.Almost all (99.7%) horse pregnancies fall in what range of lengths?What percent of horse pregnancies are longer than 339 days? Show your work.Use Table A to find the proportion of observations from a standard Normal distribution that falls in each of the following regions. In each case sketch a standard Normal curve and shade the area representing the regionz ≤ -2.25z ≥ -2.25z > 1.77-2.25 < z < 1.77Working BackwardsFind the number z at the 80th percentile of a standard Normal DistributionFind the number z such that 35% of all observations from a standard Normal distribution are greater than z.Low birth weight babies Researchers in Norway analyzed data on the birth weights of 400,000 newborns over a 6-year period. The distribution of birth weights is Normal with a mean of 3668 grams and a standard deviation of 511 grams. Babies that weigh less than 2500 grams at birth are classified as “low birth weight.”What percent of babies will be identified as low birth weight? Show your work.Fruit fly thorax lengths Here are the lengths in millimeters of the thorax for 49 male fruit flies..64.64.64.68.68.68.72.72.72.72.74.76.76.76.76.76.76.76.76.78.80.80.80.80.80.82.82.84.84.84.84.84.84.84.84.84.84.88.88.88.88.88.88.88.88.92.92.92.94Are these data approximately normal distributed? Give appropriate graphical evidence to support your answer. (Hint: histogram)The risk of Obesity A study observes a large group of people over a 10-year period, the goal is to see if overweight and obese people are more likely to die during the study than people who weigh less. Such studies can be misleading, because obese people are more likely to be inactive and poor.What are the explanatory and response variables in the study?If the study finds a strong association between these variables, can we conclude that increased weight causes greater risk of dying? Why or why not?Born to be old? Is there a relationship between the gestation period (time from conception to birth) of an animal and its average life span? The figure shows a scatterplot of the gestational period and average life span for 43 species of animals. Describe the direction, form, and strength of the scatterplot.Three “unusual” points are labeled on the graph: Point A is for the hippopotamus, Point B is for the giraffe, and Point C is for the Asian elephant. In what way is each of these animals “unusual”?Penguins Diving A study of king penguins looks for a relationship between how deep the penguins dive to seek food and how long they stay under water. For all but the shallowest dives, there is a linear relationship that is different for different penguins. The study gives a scatter plot for one penguin titled “the relation of Dive Duration (y) to Depth (x).” Duration y is measured in minutes and depth x is in meters. The report then says “the regression equation for this bird is y=2.69+ 0.0138x ”What is the slope of the regression line? Explain in specific language what this values says about this penguins dive.According to the regression line, how long does a typical dive to a depth of 200 meters last?Does the y intercept of the regression line make any sense? If so, interpret it. If not, explain why not.Stats Teachers’ Cars A random sample of AP Statistics teachers was asked to report the age (in years) and mileage of their primary vehicles. A scatterplot of the data, a least-squares regression printout, and a residual plot are provided below:Give the equation of the least-squares regression line for these data. Identify any variables you use.One teacher reported that her 6-year old car had 65,000 miles on it. Find the residual.Interpret the slope of the line in contextWhat’s the correlation between car age and mileage? Interpret this value in context.How well does the regression line fit the data? Late Bloomers? Japanese cherry trees tend to blossom early when spring weather is warm and later when spring weather is cool. Here are some data on the average March temperature (in degrees Celsius) and the day in April when the first cherry blossom appeared over a 24-year period.Temperature (Celsius) to first bloom:4.05.43.22.64.24.74.94.04.9Days in April:1481119141414219Temperature (Celsius) to first bloom:3.84.05.14.31.53.73.84.54.1Days in April:141311132817191017Temperature (Celsius) to first bloom:6.16.25.15.04.64.0Days in April:33116911Make a well labeled scatterplot in your calculator that’s suitable for predicting when the cherry trees will bloom from the temperature. The on your paper describe the direction, form, and strength of the relationship. Use technology to find the equation of the least-squares regression line. Interpret the slope and y intercept of the line in this setting.The average March temperature this year was 3.5 degrees Celsius. When would you predict that the first cherry blossom would appear? Show your method clearly. Find the residual for the year when the average March temperature was 4.5 degrees Celsius. Show your work.Use technology to find and interpret r2 in this setting.What’s my Grade? In Professor Friedman’s economics course, the correlation between the student’s total scores prior to the final examination and their final examination score is r = 0.6. The pre-exam totals for all students in the course have a mean of 280 and standard deviation of 30. The final-exam scores have a mean of 75 and standard deviation of 8. Professor Friedman has lost Julie’s final exam but knows that her total before the exam was 300. He decides to predict her final-exam score from her pre-exam total.(Hint: you need two equations to find b (slope) and a (y-intercept). Plug in values)Find the equation for the appropriate least-squares regression line for Professor Friedman’s prediction. Interpret the slope of this line in context.Use regression line to predict Julie’s final exam.Julie doesn’t think this method accurately predicts how well she did on the final exam. Determine r2 . Use this result to argue that her actual score could have been much higher (or much lower) than the predicted value. When it Rains it Pours The figure below plots the record high yearly precipitation in each state against that states record high 24 precipitation. Hawaii is a high outlier, with a record high yearly record of 704.83 inches of rain recorded at Kukui in 1982The correlation for all 50 states in the figure is 0.408. If we leave out Hawaii, would the correlation increase, decrease, or stay the same? Explain.Two least-squares lines are shown on the graph. One was calculated using all 50 states, and the other omits Hawaii. Which line is which? ExplainOntario Health Survey The ministry of Health in the province of Ontario, Canada, wants to know whether the national health care system is achieving its goals in the province. Much information about health care comes from patient records, but that source doesn’t allow us to compare people who use health services with those who don’t. So the Ministry of Health conducts the Ontario Health Survey, which interviewed a random sample of 61,239 people who live in the province of Ontario.What is the population of this sample survey? What is the sample?The survey found that 76% of males and 86% of females in the sample had visited a general practitioner at least once in the past year. Do you think these estimates are close to the truth about the entire population? Why?Bad Sampling? A large high school wants to gather student opinion about parking for students on campus. It isn’t practical to contact all students.Give an example of a way to choose a voluntary response sample of students. Explain how this method could lead to bias.Give an example of a way to choose a convenience sample of students? Explain how this method could lead to bias.Drug Testing A baseball team regularly conducts random drug tests on its players. The 25 members of the team are listed below.BataMarfGongAgarwalMoserMichiBlackmanJeffriesAndrewsMusselmanFoderBeanFrankBaerPawnicaBebowBooneyLeeBergerPetrucelliEeeBedooChenBrockmanWilsonExplain how you would chose an Simple Random Sample.Polling the faculty A researcher wants to study the attitudes of college faculty members about the work habits of entering freshmen. These attitudes appear to differ depending on the type of college. The American Association of University Professors classifies colleges as follows:Class I: Offer doctorate degree and award at least 15 per yearClass IIA: Award degrees above the bachelor’s but are not in Class IClass IIB: Award no degrees beyond the bachelor’sClass III: Two year collegesThe researcher would like to survey about 200 faculty members. Would you recommend a simple random sample, stratified random sample, or cluster sample? Justify your answer and then write a complete write up on how to carry out such a sample. Been to the movies? An opinion poll calls 2000 randomly chosen residential telephone numbers, then ask to speak with an adult member of the household. The interviewer asks, “How many movies have you watched in a movie theater in the past 12 months?” In all, 1,131 people responded.Identify a potential source of bias related to the question being asked. Suggest a change that would help fix this problem. Identify a potential source of bias in this survey that is not related to question wording. Suggest a change that would help fix this problem.Are Anesthetics safe? The National Halothane Study was a major investigation of the safety of anesthetics used in surgery. Records of over 850,000 operations performed in 34 major hospitals showed the following death rates for four common anesthetics.Anesthetic:ABCDDeath Rate:1.7%1.7%3.4%1.9%There seems to be a clear association between the anesthetic used and the death rate of patients. Anesthetic C appears to be dangerous.Explain why we call the National Halothane Study an observational study rather than an experiment, even though it compared the results of using different anesthetics in actual survey.When the study looked at lurking variables that are related to a doctors choice of anesthetics it found that anesthetic C was not causing extra deaths. Describe how a lurking variable could lead to confounding in this setting.Attitude towards Homeless people Negative attitudes toward poor people are common. Are attitudes more negative when a person is homeless? To find out, read to subjects a description of a poor person. There are two versions. One begins Jim is a 30-year-old single man. He is currently living in a small single-room apartment. The other description begins Jim is a 30-year-old single man. He is currently homeless and lives in a shelter for homeless people.After reading the description, ask subjects what they believe about Jim and what they think should be done to help him. The subjects are 544 adults interviewed by telephone. Outline, by a diagram, a completely randomized design for this experiment. Then explain how you would implement your design in detail.An herb for depression? Does the herb Saint-John’s-Wort relieve major depression? Here is an excerpt from the report of a study of this issue, “Design: Randomized, double-blind-, placebo-controlled clinical trial” The study concluded that the herb is no more effective than a placebo.How did the design of this experiment account for the placebo effect?Why is double-blind design a good idea in this setting?Vitamin C for Marathon Runners An ultramarathon, as you might guess, is a footrace longer than the 26.2 miles of a marathon. Runners commonly develop respiratory infections after an ultramarathon. Will taking 600 milligrams of vitamin C daily reduce those infections? Researchers randomly assigned ultramarathon runners to receive either vitamin C or a placebo. Separately, they also randomly assigned these treatments to a group of non-runners the same age as the runners. All subjects were watched for 14 days after the big race to see if infections developed.What type of experimental design did the researchers use? Why is this design preferable in this setting? Give one reason.Explain the purpose of the random assignment.The report of the study said, “sixty-eight percent of the runners in the placebo group reported the development of symptoms of upper respiratory tract infection after the race; this was significantly more than that reported by the vitamin C-Supplemented group (33%)” explain to someone who knows no statistics why “significantly more” means there is good reason to think that vitamin C works. How long did I work? A psychologist wants to know if the difficulty of a task influences our estimate of how long we spend working at it. She designs two sets of mazes that subjects can work through on a computer. One set has easy mazes and the other has hard mazes. Subjects work until told to stop (after about 6 minutes, but subjects do not know this). They are then asked to estimate how long they worked. The psychologist has 30 students available to serve as subjects. Describe the experiment using a completely randomized design to learn the effect of difficulty on estimated time.Describe a matched pairs experimental design using the same 30 subjects.Which design would be more likely to detect a difference in the effects of the treatments? Explain.Deceiving subjects? Students sign up to be subjects in a psychology experiment. When they arrive, they are told that interviewers are running late and are taken to a waiting room. The experimenters then stage a theft of a valuable object left in the waiting room. Some subjects are alone with the thief, and others are in pairs- these are the treatments being compared. Will the subjects report the theft?The students had agreed to take part in an unspecified study, and the true nature of the experiment is explained to them afterwards. Does this meet the requirement of informed consent? Explain.Does this study allow for inference about population, inference about cause and effect, both, or neither? Justify your answer.Roulette Gamblers often look at the electronic display in casinos and when they see the past several numbers that have come up were the color black, they say, “red is definitely due.” What’s wrong with that statement in a statistical stand point? Click it or ticket from police records, it has been determined that 15% of drivers stopped for routine license checks are not wearing seat belts. If a police officer stops 10 vehicles, “how likely is it that two consecutive won’t be wearing their seat belts?” Describe the design of a simulation to estimate this probability. Explain clearly.Weird dice nonstandard dice can produce interesting distributions of outcomes. Suppose you have two balanced, six sided dice. Die A has faces with 2, 2, 2 ,2, 6, and 6 spots. Due B has faces with 1, 1, 1, 5, 5, and 5 spots. Imagine that you roll both dice at the same time. Find a probability model for the difference (die A – die B) in the total number of spots on the up-facesProbability talk According to the “book of odds”, the probability that a randomly selected U.S. adult usually eats breakfast is 0.61Explain what probability 0.61 means in this settingWhy doesn’t the probability say that if 100 U.S adults are chosen at random, exactly 61 of them usually eat breakfast?Cholesterol Chose an American adult at Random. Define two events:A = the person has a cholesterol level of 240 milligrams per deciliter of blood (mg/dl) or above. (High cholesterol)B= The person has a cholesterol level of 200 to 239 (borderline high cholesterol)According to the American Heart Association: P(A) = 0.16 and P(B) = 0.29Explain why events A and B are mutually exclusive.If C is the event that a person chosen has normal cholesterol (below 200 mg/dl) what's P(C)Intersections and Unions Consider the two:If I said “A and B” would that be (intersection, union) and the operation implies (addition, multiplication). Write the notation.If I said “A or B” would that be (intersection, union) and the operation implies (addition, multiplication). Write the notation.Getting into College Jamie wants to go to college. So she applied to both Chico and Cal Poly. According to her counselor, the probability that Chico will admit her is 0.70 and the probability that Cal Poly will admit her is 0.60, and the probability that both will admit her is 0.40Make a Venn diagram to model this chance process.What is the probability that neither university admits her?What’s the probability that she gets into at least one of the two schools? Use the general addition rule to confirm that your answer is correct.Desktops and Laptops A computer company makes desktops and laptop computers at factories in three states- California, Texas, and New York. The California factory produces 40% of the company’s computers, the Texas factory makes 25%, and the remaining 35% are manufactured in New York. Of the computers made in California, 75% are laptops. Of those made in Texas and New York, 70% and 50% respectively, are laptops. All computers are first shipped to a distribution center in Missouri before being sent out to stores. Suppose we select a computer at random from the distribution center.Construct a tree diagram to represent the situation.Find the probability that it’s a laptop from California. Show your work.Find the probability that a desktop will be produced. Show your work.Find the probability that a lap top is produced in Texas or California. Show your work.Based on the fact that the device comes from New York, what’s the probability it’s a desktop. Show your work.For problems 36 and 37 compute the conditional probability PAB, as well as PBAPA=0.7, PB=0.4, PA∩B=0.25PA=0.45, PB=0.8, PA∩B=0.3You work at Mike’s Pizza shop. You have the following information about the 7 pizzas in the oven in the below two way table.ThickThinTotalMushroom224No Mushroom112Total347Are the events “getting a thick crust pizza” and “getting a pizza with mushrooms” independent? Explain and show your work.You add an eighth pizza to the oven. This pizza as thick crust with only cheese. Now are the events “getting a thick crust pizza” and “getting a pizza with mushrooms” independent? Explain and show your work. The “random walk” theory of stock prices hole that price movements in disjoint time periods are independent of each other. Suppose that we record only whether the price is up or down each year, and that the probability that our portfolio rises in price in any one year is 0.65. (This portfolio is approximately correct for a portfolio containing equal dollar amounts of all common stocks listed on the New York Stock Exchange.)What is the probability that our portfolio goes up for three consecutive years? A company has developed a drug to detect steroid use by athletes. Suppose that a drug test will be used in a populations of athletes in which 10% have actually taken steroids. The test is accurate 95% of the time when an athlete has taken steroids. It is 97% accurate when an athlete hasn’t taken steroids. Let’s choose an athlete at random and administer the drug.Make a tree diagram showing the sample space of this chance process.What’s the probability that a randomly selected athlete tests positive? Show your work. Knees Patients receiving artificial knees often experience pain after surgery. The pain is measured on a subjective scale with possible values of 1 (low) to 5 (high). Let X be the pain score for a randomly selected patient. The following table gives part of the probability distribution for XValue:12345Probability:0.10.20.30.3??Find P(X=5)Is pain score a discrete or continuous random variable? Explain.Find P(X≤2) is this the same as P(X<2)? pute the expected value and standard deviation of X. Show your work.A Glass Act In a process for manufacturing glassware, glass stems are sealed by heating them in a flame. The temperature of the flame can be adjusted to one of five different settings. Here is the probability distribution of the flame temperature X (in degrees Celsius) for a randomly chosen glass:Temperature (degrees):540545550555560Probability:0.10.250.30.250.1Is temperature a discrete or continuous random variable? Explain.How is P(X<540) related to P(X≤540)? Explain.The target temperature is 550 degrees Celsius. What are the mean and standard deviation of the number of degrees off target, D = X -550?A manager asks for results in degrees Fahrenheit. The conversion of X into degrees Fahrenheit is given by Y=95X+32. What are the mean and standard deviation of the temperature of the flame given in Fahrenheit scale? Use Y as your new variable. Keno In a game of 4-spot Keno, the player picks 4 numbers from 1 to 80. The casino randomly selects 20 winning numbers from 1-80. The table below shows the possible outcomes of the game and their probabilities, along with the amount of money (Payout) that the player wins for a $1 bet. If X= the payout for a single $1 bet, you can check that ?x=$0.70 and σx=$6.58Matches:01234Probability:0.3080.4330.2130.0430.003Payout:$0$0$1$3$120Interpret the values of ?x and σx in context.Jerry places a single $5 bet on 4-spot Keno. Find the expected value and the standard deviation of his winnings.In exercise 44 and 45 refer to the following setting. According to the Mars candy company, 20% of its plain M&M’s candies are orange. Assume that the company’s claim is true. Suppose that you reach into a large bag of plain M&M’s (without looking) and pull out 8 candies. Let X= the number of orange candies you get. INDEPENDENCE: 10 % condition satisfies this (we will talk about this much more 2nd semester)Orange M&M’s Explain why X is a binomial variable.Find and interpret the expected value of X.Find and interpret the standard deviation of X.Orange M&M’sWould you be surprised if none of the candies were orange? Compute an appropriate probability to support your answer.How surprising would it be to get 5 or more orange candies? Compute an appropriate probability to support your answer.Sushi Roulette In the Japanese game show Sushi Roulette, the contestants spins a large wheel that’s divided into 12 equal sections. Nine of the sections have a sushi roll, and three have a “wasabi bomb.” When the wheel stops, the contestant must eat whatever food is on that section. To win the game, the contestant must eat one wasabi bomb. Find the probability that it takes 3 or more spins for the contestant to get a wasabi bomb. Show your method clearly. ................
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