Unit 08A (Chapter 11): Probability and Data Analysis

Note to CCSD Pre-Algebra Teachers: In the curriculum engine, 3 rd quarter CCSD syllabus objectives include the last section of Chapter 6 (probability and odds) and then address Chapter 11 objectives. Also included in this quarter is the rollout of the Common Core State Standards for grade 8 in the domain of Statistics and Probability. Since this CCSS rollout will only be field tested on this year s CRT, and all the Nevada State Standards remain in the curriculum and have the potential to appear on this year s CRT, this unit of study will be divided into two parts: Unit 08A (Chapter 11): Probability and Data Analysis (addressing CCSD syllabus objectives and Nevada state standards) Unit TBA: Data and Prediction (addressing the CCSS standards) Unit 8A (Chapter 11) must be taught before the CRT. Unit TBA will not appear until much later in the school year. Unit 08A (Chapter 11): Probability and Data Analysis Specification Sheet McDougal Littell CCSD Syllabus Objective Reference The student will: 2.15 Find the probability of an event. 6.7 2.26 Differentiate between the probability of an event and the odds of an event. 6.7 5.1 Make stem-and-leaf plots, box-and-whisker plots, and histograms. 11.1, 11.2 5.10 Apply appropriate measures of data distribution, using interquartile range and central tendency. Pg 39-40 5.4 Formulate inferences and predictions through interpolation and extrapolation of data to solve practical problems. 11.3,11.5 5.11 Evaluate statistical arguments that are based on data analysis for accuracy and validity. 11.3,11.5 5.3 Formulate questions that will guide the collection of data. 11.4 5.2 Design data analysis projects. Pg 646-647 Unit (TBA) Data and Prediction 8.SP.1-1 8.SP.1-2 8.SP.2 CCSS Standards Specification Sheet The student will: Construct scatter plots for bivariate measurement data to investigate patterns of association between two quantities Interpret scatter plots for bivariate measurement data. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, McDougal Littell Reference McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 1 of 13

8.SP.3 8.SP.4-1 8.SP.4-2 8.SP.4-3 informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct a two-way table summarizing data on two categorical variables collected from the same subjects. Interpret a two-way table summarizing data on two categorical variables including using relative frequencies calculated for rows or columns to describe possible association between the two variables McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 2 of 13

Pre-Algebra 8 Notes Chapter 11: Probability and Data Analysis Probability Syllabus Objective: (2.15) The student will find the probability of an event. Probability is the measure of how likely an event is to occur. They are written as fractions or decimals from 0 to 1. Probability may be written as a percent, 0% to 100%. The higher the probability, the more likely an event is to happen. For instance, an event with a probability of 0 will never happen. If you have a probability of 100%, the event will always happen. An event with a probability of 1 2 or 50% has the same chance of happening as not happening. Example: How likely is it that a coin tossed will come up heads? This means that there is as likely a chance of heads as not heads. In other words, a probability of 1 or 0.5 or 50%. 2 Example: The weather report gives a 75% chance of rain for tomorrow. This means that there is a likely chance of rain (75%) and an unlikely chance of no rain (25%). In other words, the probability of rain is 3 or 0.75 or 75%. 4 Outcomes are the possible results of an experiment. Example: Tossing a coin, the possible outcomes (results) are a head or a tail. Theoretical probability is based on knowing all the equally likely outcomes of an experiment, and it is defined as a ratio of the number of favorable outcomes to the number of possible outcomes. Mathematically, we write: probability = number of favorable outcomes number of possible outcomes or success probability = success + failure McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 3 of 13

Example: Suppose you pick a marble from a hat that contains three red, two yellow and one blue marble. What is the probability you draw a yellow marble? # of yellow marbles P = total number of marbles 2 1 P = or 6 3 The probability found in the above example is an example of theoretical probability. Experimental probability is based on repeated trials of an experiment. Example: In the last thirty days, there were 7 cloudy days. What is the experimental probability that tomorrow will be cloudy? 7 P = 30 Note: If one event does not depend on another, probability is not changed by previous events. In other words, if you toss a fair coin 10 times, and it has not landed heads-up yet, the probability of it landing heads up on the next toss is still ½ or 50%. Probability of Two Events You may want to find the probability that two events will both occur. To do this, you first have to determine whether the two events are dependent of independent. When the outcome of one event does not affect the outcome of another event, the two events are independent. You can use the formula: Probability (A and B) = Probability (A) Probability (B). Example: You have the following cards left in a deck: 3, 4, 4, 5, 6, 6, 7, 8. Suppose you pick a card replace it in the deck, shuffle the deck, and pick a second card. What is the probability that you will pick 6, then another 6? If you pick 6, then replace the card in the deck, the first pick doesn t affect your second pick, so the two events are independent. Probability (6, then = P(6) P(6) 2 2 = 8 8 4 1 = or 64 16 6) McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 4 of 13

When the outcome of one event affects the otcome of another event, the two events are dependent. You can use the formula: Probability (A and B) = Probability (A) Probability (B, given A). Example: You have the following playing cards left in a deck: 2, 3, 3, 4, 4, 6, 8, 9, 10. If you choose two cards at random, what is the probability you will pick 3 then 8? Since choosing 3 first will change the number of cards that are left to choose from, the two choices are dependent events. Probability (3, then 8) = P(3) P(8, given 3) 2 1 = number of 3' s P(3) = 9 8 number of cards 2 1 = or # of 8' sleft after first pick 72 36 P(8, given 3) = # of cardsleft after first pick Odds Syllabus Objective: (2.26) The student will differentiate between the probability of an event and the odds of an event. Odds: the ratio of favorable outcomes to the number of unfavorable outcomes, when all outcomes are equally likely. Odds in favor = Odds against = Number of favorable outcomes Number of unfavorable outcomes Number of unfavorable outcomes Number of favorable outcomes Example: Suppose you pick a marble from a hat that contains three red, two yellow and one blue marble. What are the odds (in favor) you draw a yellow marble? 2 of yellow marbles 2 1 P = = = or 1:2 # of marbles not yellow 4 2 Teachers should also address that one can convert from odds to probability, and vice versa. For example, if the odds of winning a game is 2 3, then the probability of winning is 2. If the 5 probability of rain is 7 10, then the odds of rain is 7 3. McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 5 of 13

Pre-Algebra, Unit 11: Data Analysis Notes Stem-and-Leaf Plots Objectives: (5.1) The student will make stem-and-leaf plots. Let s use the following test scores to construct a stem and leaf plot. 82, 97, 70, 72, 83, 75, 76, 84, 76, 88, 80, 81, 81, 82, 82 We first determine how the stems will be defined. In our case, the stem will represent the tens column in the scores, the leaf will be represented by the ones column. When we present our information, it will be in two parts, the stem and the leaf. Let s say I had this: 5 7 4. The way I would read that is by knowing the stem represents fifty, and the leaf has two scores, 7 and 4. Reading that information, I have a 57 and a 54. Knowing that, let s arrange our data in a stem-and-leaf plot. Knowing our lowest score is in the 70 s and the highest is in the 90 s, our stem will consist of 7, 8, and 9. Usually, the smaller stems are placed on top. You can make the decision for yourself. Another decision you can make is whether or not you put the scores in order in the leaf portion. As you can see, I didn t. 7 0 2 5 6 6 8 2 3 4 8 0 1 1 2 2 9 7 Notice that leaf part of the graph did not have to be in any particular order. So a person reading this plot would know the scores are 70, 72, 75, 76, 76, 82, 83, 84, 88, 80, 81, 81, 82, 82 and 97. What could be easier? Histogram Objectives: (5.1) The student will make histograms. A histogram is made up of adjoining vertical rectangles or bars. If we rotated the last stem-andleaf graph 90 degrees and made the rectangles as high as the leaf portion, we would have a histogram. A histogram looks like a bar graph, except the rectangles are connected. Let s actually do a problem using the information from the previous example. McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 6 of 13

A histogram would typically identify what you are talking about on the horizontal axis. The vertical axis describes the frequency of those observations. One problem you might encounter on a histogram is when data falls on the line that divides two rectangles. In which rectangle do you count the data? Another problem is the width of the rectangles: how wide do you want them? Both of these problems are easily overcome. To determine the width, first find the range, which is the difference in the largest score and the smallest. 70, 72, 75, 76, 76, 82, 83, 84, 88, 80, 81, 81, 82, 82 and 97 Using the data from the example we have: 97 70 = 27 If you wanted three categories, you divide 27 by three; then each width would be about nine. If you wanted four categories, you d divide 27 by 4; then the width would be a little bigger than 6. It s your decision. No big deal. That takes care of the width problem. Now what about if something falls on a line that separates the rectangles? Do we count it in the left or right rectangle? Well, we just won t let that happen. We ll expand the range by one half then no score can fall on a line. Don t you just love how easy that was to take care of? So, I m deciding to have four groups, the width is a little more than 6 I ll say seven. And I m going to begin at 69.5 rather than 70. That should result in all my data falling within a rectangle.. Let s see what it looks like. Box-and-Whisker Plot Objectives: (5.1) The student will make box-and-whisker plots. (5.10) The student will apply appropriate measures of data distribution, using interquartile range. Let s take a look at what might be a way of giving notes to your students to help them learn the steps for creating a box-and-whisker plot. See the next page! McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 7 of 13

Making a Box & Whisker Plot You try: 10 12 8 14 16 16 11 13 11 15 8 Steps Ex: 5 10 7 9 8 6 11 1. Arrange data in increasing order. (minimum value & 5 6 7 8 9 10 11 maximum value are the endpoints) 2. Find median of the entire list (median value) 5 6 7 8 9 10 11 a. If there is a number in the list that is the middle term, circle it and draw a line thru it. (median) b. If there is not a number that is in the middle, draw a line between the two numbers. (median is the number halfway between the two numbers) 3. Look at the bottom half of the numbers. Find the median of the bottom half of numbers (lower quartile) (same as #2 above) 4. Look at the top half of the numbers. Find the median of the bottom half of numbers (upper quartile) (same as #2 above) 5. Draw a number line that will cover the range of data (Evenly spaced marks) 6. Slightly above the number line place dots at the following points: minimum, lower quartile, median, upper quartile, and maximum 8 is the median 5 6 7 8 9 10 11 Does not apply to this problem 5 6 7 9 10 11 4 6 8 10 12 14 16 4 6 8 10 12 14 16 McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 8 of 13

7. Draw a box with side borders being the lower and upper quartiles. Draw two lines, one from each side of the pox connecting the minimum point on one side and the maximum point on the other side. Draw a vertical line at the median point from the top to the bottom of the box. 4 6 8 10 12 14 16 minimum Lower Quartile median Upper Quartile maximum 4 6 8 10 12 14 16 The data in the box represents the Interquartile Range IQR, the average, the middle 50%. The whisker on the left represents the bottom quartile, the bottom 25%; the whisker on the right represents the top 25%. The difference between the upper and lower quartiles is called the interquartile range (IQR). A statistic useful for identifying extremely large or small values of data is called an outlier. An outlier is commonly defined as any value of the data that lies more than 1.5 IQR units below the lower quartile or more than 1.5 IQR units above the upper quartile. In our example the lower quartile was at 6, the upper at 10. Using that the IQR =10 6 = 4. Multiplying that by 1.5, we have ( 1.5)( 4) = 6 Therefore, any score below 6 6 = 0 is an outlier, as is any score above 10 + 6 = 16. There are no points below 0, so we are OK on the left. There are no points greater than 16, so we are OK on the right. This would be an ideal place to use technology. Next are the instructions for drawing a box-and-whisker plot on the TI84. The examples will address outliers. McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 9 of 13

Entering Data and Drawing a Box-and-Whisker plot on the TI84 modified 1. STAT 2. EDIT regular 3. Enter the numbers in List 1 (L 1 ) or List 2 (L 2 ) 4. Return to the home screen 5. STAT PLOT 6. Turn Stat Plot 1 on and select the type of boxplot (modified or regular) 7. ZOOM 8. ZoomStat (9) 9. GRAPH Modified Regular The Ti 84 graphing calculator may indicate whether a box-and-whisker plot includes outliers. One setting on the graphing calculator gives the regular box-and-whisker plot which uses all numbers, so the furthest outliers are shown as being the endpoints of the whiskers Another calculator setting (modified) gives the box-and-whisker plot with the outliers specially marked (in this case, with a simulation of an open dot), and the whiskers going only as far as the highest and lowest values that aren't outliers: Find the outliers and extreme values, if any, for the following data set, and draw the box-and-whisker plot. Mark any outliers with an asterisk and any extreme values with an open dot. 20, 21, 21, 23, 23, 24, 25, 25, 26, 27, 29, 33, 40 To find the outliers and extreme values, I first have to find the IQR. Since there are thirteen values in the list, the median is the seventh value, so Q 2 = 25. The first half of the list is 20, 21, 21, 23, 23, 24, so Q 1 = 22; the second half is 25, 26, 27, 29, 33, 40 so Q 3 = 28. Then IQR = 28 22 = 6. The outliers will be any values below 22 1.5 6 = 22 9 = 13 or above 28 + 1.5 6 = 28 + 9 = 37. The extreme values will be those below 22 3 6 = 22 18 = 4 or above 28 + 3 6 = 28 + 18 = 46 Another example: L 2 =21, 23, 24, 25, 29, 33, 49 So I have an outlier at 49 but no extreme values, I won't have a top whisker because Q 3 is also the highest non-outlier, and my plot looks like this: McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 10 of 13

Measures of Central Tendency Objectives: (5.10) The student will apply measures of data distribution using central tendency. 3 Measures of Central Tendency 1. Mean 2. Median 3. Mode The mean is the one you are probably most familiar with; it s the one often used in school for grades. To find the mean, you simply add all the scores and divide by the number of scores. In other words, if you had a 70, 80, and 90 on three tests, you d add those and divide by three. The mean is 80. Your average is 80. The median, often used in finance, is the middle score when the data is listed in either ascending or descending order. If there is no middle score, then you take the two middle scores, add them and divide by 2 (find the average of the two scores) Example: Find the median of 72, 65, 93, 85, and 55. Rewriting in order, I have 55, 65, 72, 85 and 93. The middle score is 72, so the median is 72. Piece of cake, right? The mode is the piece of information that appears most frequently. Let s look at some data: 55, 64, 64, 76, 78, 81, 81, 81, and 92. What scores appears most often? If you said 81, you just named the mode. You ve used the mode quite often before. If you have ever described the average weight of a particular population, the average height, shoe size, shirt size, the number of points scored in a particular type of game those are all examples of you using the mode. Note: Don t forget about an outlier. When a piece of data is much bigger or smaller than the rest, it is called an outlier. It can move the mean away from the main group of data toward the outlier. For instance, if a book store had sales of books (number of books sold) for one week of 105, 96, 90, 108 and 116, the mean would be 105 + 96 + 90 + 108 + 116 = 515, and 515 5 = 103. However, if on the next day, the book store only sold 7 books, the new mean would be 87. One day with very low sales brought the mean sales per day down by nearly 20 books! It made the mean smaller than any of the other pieces of data, so the mean no longer describes a typical day of sales. Interpreting Data Objectives: (5.4) The student will formulate inferences and predictions through interpolation and extrapolation of data to solve practical problems. (5.11) The student will evaluate statistical arguments that are based on data analysis for accuracy and validity. When looking at results, you must consider several questions like: Is the data display misleading? What is the margin of error? Are the conclusions supported by the data? McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 11 of 13

Margin of error of a random sample defines an interval centered on the sample percent in which the population percent is most likely to lie. Example: A sample percent of 27% has a margin of error of ±4%. Find the interval in which the population percent is most likely to lie. 27% 4% = 23% and 27% + 4% = 31%. The interval is between 23% and 31%. Predictions can be made about populations. If p % of a sample gives a particular response and the samples is representative of the population, then p% (# of people in population ) = predicted # of people in the population giving the response Example: A survey of 300 randomly selected cat owners finds that 120 cat owners prefer Brand C cat food. Predict how many owners in a town of 2000 cat owners prefer Brand C cat food. The percent of cat owners in the sample that preferred Brand C is 120 = 40% ; 40% of 2000 = 800 cat owners 300 Translating the objectives for interpreting data into instruction can also be done by looking at samples. McDougal Littell s Pre-Algebra Book, sections 11.3 to 11.5, contains many such examples and problems. Here are some other sample problems with which to begin: Example 1: Which of the following graphs most accurately depicts the hourly wages earned with respect to time worked? A. B. C. D. Earnings Earnings Earnings Earnings Hours Worked Hours Worked Hours Worked Hours Worked Discussion: Students should recognize that if one is paid by the hour, then only Graph C could illustrate it. A person would not earn money at no (zero) hours as shown in graph A. Looking at the other graphs, you could rule out B as a person would not earn the same amount for different amounts of hours worked. Graph D shows a person earning less money as the person works more hours. McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 12 of 13

Example 2: The graphs below show the numbers of baskets made by Player A and Player B during 5 basketball practices. Each player takes 100 practice shots during each practice. According to the graphs, who was more successful at making baskets? Player A Player B # of Baskets 100 80 60 40 20 # of Baskets 100 80 80 78 60 76 40 74 20 72 A. Player A did much better. B. Player B did much better. C. Their scores appear to be about the same. D. More information is needed. 0 0 1 2 3 4 5 0 0 1 2 3 4 5 Practice Session Practice Session Discussion: In this problem, there will be students who think Player B has scored the most baskets because of the steepness of the linear segments. Some students will think Player A scored the most baskets because his line segments looks consistently higher. They need to look carefully at the vertical scaling and note that Player A started about 70-71 and ended at 80, the same as Player B. (Answer is C.) Example 3: Why is this graph misleading? 50 Season Attendance (in thousands) The heights of the balls are used to represent the number of spectators. # of spectators (thousands) 40 30 20 10 However, the area of the balls distorts the comparison. The attendance for both sports doubled in the 10 year period; the size of the basketball makes it look like that increase may have been a lot more. 0 0 1990 2000 1990 2000 Year McDougal Littell, Chapter 6, Section 7 Pre-Algebra, Chapter 11: Probability and Data Analysis Page 13 of 13