Think: Know where you re headed in your analysis and why. Know your goal. This helps you understand what you re doing.

Transcription

1 AP Statistics Summer Packet (Summer 2014) Statistics is the study of variation. That s what we re going to be doing for the entire school year.we re going to be studying variation. If everything in this world always happened the way it was supposed to, this would be a very boring world and there would be no reason for the study of Statistics. We re going to be studying data. Data can take two forms Categorical and Quantitative. The vast amount of our year will be devoted to the study of quantitative data. This is data that you can put into your calculator and perform numeric analysis to. You ll need a graphing calculator with statistical capabilities. Our textbook (like most textbooks) uses the TI- 83 and TI-84 models for demonstrations. The TI-89 can be used, but the calculator commands have subtle and not so subtle differences. The Casio FX-9750 is actually more closely in alignment to the TI-83 and 84 than the 89. It s also cheaper. Categorical data, on the other hand, can be analyzed with the use of a basic function calculator. Categorical data, by its natural form, does not lend itself to numeric analysis (such as mean, median, standard deviation, correlation, quartiles, skewness..). Categorical data is studied in Chapter 3 in or textbook and then much later on toward the end of the school year. The purpose of this summer packet is to teach Chapter 3-Displaying and Describing Categorical Data. Background The first two chapters of the textbook try to set the stage for the rest of the year. The study of Statistics is the study of variation. This will be a very different type of math course when compared to other math courses you ve taken. You will be writing more sentences in this course than you will be solving equations. This course is all about coming up with an answer and then explaining the answer. Justifying the answer. Supporting the answer. Defending the procedure you used to get the answer. The textbook introduces a three-step method to performing a statistical analysis: Think Show Tell Think: Know where you re headed in your analysis and why. Know your goal. This helps you understand what you re doing. Show: The actual mechanics of what you re doing. If you have to make a graph, make the graph. If you have to do a certain set of calculations, do the calculations. This is what students find to be easy. Tell: Explain what you ve learned. Until you ve explained your results so that someone else can understand your conclusions, the job is not job. This is what you ll have to learn more than anything else. In all of your other math classes, the job has been done once you get the answer. In this class, the calculator often times does the work of getting the answer. Your job is to explain what the answer means. On to Chapter 3 The three rules of data analysis. 1. Make a picture 2. Make a picture 3. Make a picture Get the point? We will be using diagrams and charts and graphs (oh my!) a lot.

2 Remember, Chapter 3 is all about Categorical data. The easiest way to consider whether data is categorical or quantitative is to put yourself into the shoes of the person gathering the data. What data are you seeking? If you re interested in polling seniors at your high school and wondering about how many colleges each senior applied to, you would naturally be gathering data that is numeric and would be capable of being analyzed in your calculator. (The mean number of colleges applied to is and the standard deviation is ) However, if you re interest in the distribution of vehicle colors in the student parking lot, your data would not be numeric. You could still poll students who drive to school, but the answer you would get to your question (what color is your car or truck?) would not take the form of a number, it would be a word (red, white, black ). That s categorical data. One word of caution, just because it s a number doesn t mean it s numeric data. If we asked 5 football players what their jersey number was and got the responses 8, 25, 41, 17 and 30 it would make no sense to do a numeric analysis and say the average jersey number is 24.2 with a standard deviation of In this case, even though the data is numeric, it s not quantitative, it s categorical. It s descriptive. Categorical data is descriptive. Other examples of numbers which really act as categorical data are Social Security and locker combinations. The analysis of categorical data is limited. That s why is doesn t warrant too many chapters in our textbook. Displaying the data can take the forms of pie charts (circle graphs), bar graphs and contingency tables. Our book uses the data relating to those who were on the Titanic. Specifically, it breaks the passengers down by 2 different analysis: Survival Status (did they survive or die) and Passenger Status (were they 1 st class, 2 nd class, 3 rd class or Crew). If we break the data down by the variable Survival Status, it would look like this: Alive Dead Total ,201 If we break the data down by the variable Passenger Status, it would look like this: First Class Second Class Third Class Crew Total ,201 We could make bar graphs and pie charts displaying these data. It would be fairly simple and fairly boring. If we made a pie chart of the variable Survival Status, it would only have 2 sections. Alive would represent 32.3% of the circle, or degrees, while dead would be the other section, 67.7% or degrees. We could do the same type of display with the other variable-passenger Status. If we chose to display using bar graphs, we could either use the actual values (marginal frequency) or the percentages (relative frequency). The relative frequency of Passenger Status is: 1 st Class 2 nd Class 3 rd Class Crew Total 14.8% 12.9% 32.1% 40.2% 100% A much more interesting analysis of the data would be obtained if we studied both variables at the same time. This is accomplished through the use of a Two-Way table, often called a Contingency Table.

3 Passenger Status 1 st Class 2 nd Class 3 rd Class Crew Total Survival Alive Status Dead Total Don t be fooled. There are only two variables in this analysis. When asked to identify the number of variables in this analysis often times I ll get the answer 6 (1 st, 2 nd, 3 rd, Crew, Alive and Dead) So, what can we do with this data? First of all, I can ask all the different types of probability questions you were asked when you studied this in the Probability unit of Algebra 2. If an individual was selected at random from this table, what s the probability that the individual. 1. P(survived) 711/2201 = P(2 nd class) 285/2201 = P(3 rd class or died) 706/ / /2201 = 1668/2201 = P(crew and survived) 212/2201 = P(survived 1 st class) what s the probability that the individual survived GIVEN the fact that he/she was in 1 st class? 203/325 = The Case for Independence Two variables are said to be independent if the occurrence or nonoccurrence of one plays no influence on the probability of occurrence or nonoccurrence of the other. Back in Algebra 2, we had this equation that read P(A and B) = P(A) * P(B). This was known as the AND rule. When you were given an example such as if P(A) = 0.6 and P(B) = 0.7 what s P(A and B) and your answer was 0.42, you were correct under the assumption that events A and B are independent. Event A plays no influence on Event B. Sometimes we have problems that go like this: If P(A) = 0.6 and P(B) = 0.7 and P(A and B) = 0.32, then clearly events A and B are not independent. Something s going on that s influencing the probabilities. If we focus on Event B which has a 0.7 chance of happening on its own, what happens when Event A enters the picture? Find P(B A) what s the probability of Event B occurring GIVEN that Event A has occurred? P(B A) = P(A and B) / P(A) 0.32 / 0.6 = So what s going on here? What have we learned? The probability of Event B happening without regard to Event A is 0.7. That was a given. We then calculated the probability of Event B happening having known that Event A occurred by using the formula P(B A) = P(A and B) / P(A) and found the answer to be We now have two different probabilities for the same event Event B. Without

4 regard to Event A, P(B) = 0.7. Taking into consideration that Event A is known to have occurred, the P(B) = The known occurrence of Event A has decreased the likelihood of Event B. It s taken the probability of Event B down from 0.7 to If Event A is known to have happened, the probability of Event B is no longer 0.7, it s Events A and B are not independent. Let s study the same question in regards to the two variables Survival Status and Passenger Status. We ve all seen the movie so we already know what we re supposed to get as an answer. Your chances of surviving did depend on your status as a passenger. Let s show it mathematically. Event A First Class Passenger P(A) = /2201 Event B Survived P(B) = /2201 Under the assumption of Independence P(A and B) is P(A) * P(B) = (0.148)(0.323) = If Passenger Class and Survival Status were independent, only 4.8% of the passengers on the ship would be 1 st Class passengers who survived. However, the real proportion of passengers who were 1 st Class and survivors was 9.2% (203/2201). Almost double! Here s another way of looking at: Only 14.8% of all passengers were 1 st Class P(1 st class) = Only 32.3% of all passengers survived P(survived) = However, if you were ticketed in 1 st class, you had a 62.8% chance of survival. P(survived 1 st Class) = 203/325 = Compare this to the overall survival rate of Again, almost double!

5 One more way of looking at it This time with a picture. The picture is that of Segmented Bar Graphs Percent Alive Dead The data has been broken down to class of passenger within each survival status. The sections within each bar graph go 1 st Class, 2 nd Class, 3 rd Class, Crew from the bottom to the top. Of those who survived: 28.6% were 1 st Class, 16.6% were 2 nd, 25% were 3 rd and 29.8% were Crew. Of those who died: 8.2% were 1 st Class, 11.2% were 2 nd, 35.4% were 3 rd and 45.2% were Crew. The fact that the Segmented Bar Graphs show a noticeable difference in their distributions tells us that Survival Status and Passenger Status are not independent. If Survival Status and Passenger Status were independent, the Segmented Bar Graphs would have no striking difference. As it is now, the Segmented Bar Graphs are definitely telling a story. If you were traveling in 1 st class, you were much more likely to survive than to die. If you did not survive, you were much more likely to be a member of the crew or traveling as a 3 rd class passenger. Here s a wrap up to this analysis if someone unfamiliar with the story of the Titanic were to ask a seemingly simple question like this What were your chances of surviving the sinking of the Titanic? After having done this analysis, your first response would be It depends what type of passenger are we talking about? Whenever the data tells us it depends, the variables are dependent.

6 Deceptive Graphs 3-D graphing DON T DO IT! It might look like it s presenting a more interesting picture of the data, however, by adding the 3 rd dimension, you can (and often do) distort the data you re trying to present. Scaling When scaling on a bar graph, make sure scales are uniform and begin at 0 on the bottom Violating the Area Principle see the next two pages Remember, the graph on the next page is attempting to display the marginal distribution of passengers : First Second Third Crew The impression being given by the graph is in conflict with the actual data. The boat representing crew is 3 or 4 times as big as the boat representing first class but the numbers don t agree. Do you see what they did? In order to keep the boat in scale, when they lengthened the horizontal axis they lengthened the vertical axis as well. That creates the distortion. Another Example AP Statistics Scores for Marshall High School The following page shows a Segmented Bar Graph for AP Stats scores from Marshall High School. There are two variables in this analysis. Can you identify them? AP Score and Year in School The question posed at the bottom of the graph, Is there an association? is another way of determining whether or not the two variables are independent. Remember, saying that they are independent would mean that your score on the AP Stats test would not be influenced by what year you are in school. The first step in the analysis is to simply look at the Segmented Bar Graph and determine whether or not any noticeable differences exist. If your answer to that is no, then you conclusion is easy it appears that AP Score and Year in School are independent (or not associated). It doesn t really matter what year you are in school because the breakdown of the AP Scores were pretty much consistent among the three classes (Soph, Jr and Sr). I don t think you re going to get off that easy on this problem. There does seem to be some noticeable differences in the breakdowns. When noticeable differences occur, then there is a lack of independence in the data. Another way of saying this would be to say AP Score and Year in School do seem to be associated. Some justifications you could use would be: 1. If you got a 5 on the test, you were probably not a Senior 2. If you got a 1 on the test, you were either a Jr. or a Sr.

7 3. If you got a 3 on the test, you were probably a Sr. 4. If you passed the test (3 or better) you were more likely to be a Soph. If the breakdowns were to occur across the board with no noticeable differences, then the only thing you could say about the graph would be the breakdown of AP Scores occur fairly evenly, making it not really relevant as to what grade in school you are. That s the definition of independence. Another Example: A two-way table is shown which records the favorite leisure activity of 50 adults: 20 men and 30 women. Favorite Leisure Activity Chart #1 the Marginal Distribution Dance Sports TV Total Charts #2-4 Relative Frequencies Men Gender Women Total Dance Sports TV Total Men Women Relative Frequency of Table (Grand Total) Total Dance Sports TV Total Men Women Relative Frequency of Rows (Gender) Total

8 Dance Sports TV Total Men Women Relative Frequency of Columns (Favorite Activity) Total Probability Questions: If one person where selected at random, find the following probabilities P(Woman) = 0.60 P(Dance) = 0.36 P(Man or Sports) = = 0.52 P(Woman and Dance) = 0.32 P(Man Dance) = 0.11 P(Man) = 0.40 P(TV Woman) = 0.27 Notice that all of these answers could be found by using one of the three relative frequency charts. The Case for Independence: Do Gender and Favorite Leisure Activity appear to be independent? Use a Segmented Bar Graph as part of your analysis and conclusion. We ll create the bar graphs based on Gender. Percent Men Women Think: I am going to create Segmented Bar Graphs for the Relative Frequencies of Gender. Show: see above Tell: Due to the fact that the Segmented Bar Graphs show a noticeable difference in the distributions of Favorite Leisure Activity for Men and Women, the variables appear to be dependent. There does appear to be an association between

9 whether you re a man or woman and what your favorite leisure activity is. For example, a woman is much more likely to respond dance than a man. Also, if a respondent is known to have answered sports, the person is much more likely to be a man. The response which was closest to being even was TV and that wasn t actually that close. Simpson s Paradox Just know this can occur. Two fighter pilots, Eagle and Striker, disagree as to who is the better pilot. They are using their First Attempt Landing Percentage as the comparison. When a pilot attempts to land on an aircraft carrier, they are either given the green light which means to continue to land or they re given the red light which means to abort the landing and go around and make another attempt. For further understanding, watch Top Gun. Their data for successful 1 st attempt landings appears below: Day Landings Night Landings Total Eagle 90 out of out of out of 120 (83%) Striker 19 out of out of out of 120 (78%) Each pilot is using 120 landings as the basis for comparison. Eagle claims he is the better pilot because his overall First Attempt Landing Percentage is higher than that of Striker s (83% vs 78%). Striker, however, feels she s the better pilot because when the data is broken down and analyzed separately, she has a higher percentage in both the Day Landings and the Night Landings comparisons. In Day Landings, she is 95% while Eagle is 90%. In Night Landings, she s 75% while he is 50%. How can she be better in each of the head-to-head comparisons yet he s better using the overall data? I ll leave that up to you to think about. Goals of this packet: Understand the difference between categorical data and quantitative data Be able to analyze, interpret and display categorical data Identify whether 2 variables are independent (not associated) as a result of examining segmented bar graphs Be familiar with the graphs of categorical data and their limitations Answer probability questions using a two-way table (contingency table) Be familiar with Simpson s Paradox Assignment: Do the following problems on a separate piece of paper

10 1. A May 2001 Gallup Poll found that many Americans believe in ghosts and other supernatural phenomena. The poll was based on telephone responses from 1012 randomly selected adults. The table shows the percentages of people who expressed belief in various phenomena. Phenomena Percent Expressing Belief Psychic healing 54 ESP 50 Ghosts 38 Astrology 28 Channeling 15 a) Is it reasonable to conclude that 66% of those polled expressed belief in either ghosts or astrology? Explain. b) Can you tell what percent of people did not believe in any of these phenomena? Explain. c) Create an appropriate display for these data. Be neat! 2. List some errors of this display. 3. A 1975 article in the magazine Science examined the graduate admissions process at Berkeley for evidence of gender bias. The table below shows the number of applicants accepted to each of four graduate programs. Males accepted Females accepted Program out of out of 108 Program out of out of 25 Program out of out of 375 Program 4 22 out of out of 341 Totals 1022 out of out of 849 Give a brief report of acceptance by gender for each program and then compare that to the overall acceptance rate by gender. You should see Simpson s Paradox in action. 4. If P(A) = 0.3 and P(B) = 0.5 and P(A and B) = 0.2 4a. The fact that P(A and B) is not simply the product of the individual probabilities tell us that events A and B are.. 4b. Explain the influence that event B is having upon event A by using the formula P(A B) = P(A and B) / P(B)

11 5. A survey of autos parked in student and staff lots at a large university classified the maker of the car according to the country of origin, as seen in the table. Driver Student Staff American Origin European Asian a) What % of all cars surveyed were foreign (non-american)? b) What % of the American cars were driven by students? c) What % of the students drove American made cars? d) What is the marginal distribution of Origin? e) What are the conditional distributions of Origin broken down by Driver? f) What are the conditional distributions of Driver broken down by Origin? g) Produce segmented bar graphs based on Driver (you ll have two bar graphs-one for Student and the other for Staff). h) Based on the graph you produced in g, do you think there is an association between the type of driver and origin of the car? Explain you answer with some facts from your analysis. 6. The table below compares what Ithaca High School students did after graduation in 1959, 1970 and Continued with further education Began their career Joined the military Other If a graduated was selected at random from this chart, find the following probabilities (round to 10 th ) a) P(joined military) b) P(graduated in 1970) c) P(military 1970) d) P(1980 began career ) e) P(graduated in 1959 or continued with further education) f) P(graduated in 1980 and other) g) Produce segmented bar graphs based on year of graduation. h) based on the graph you produced in g, do you think year of graduation and post high school actions are independent? Justify your answer with some facts from your analysis.