ANALYZING BIVARIATE DATA

Analyzing bivariate data 1 ANALYZING BIVARIATE DATA Lesson 1: Creating frequency tables LESSON 1: OPENER There are two types of data: categorical and numerical. Numerical data provide numeric measures for a population. The numbers of text messages each student sends per week are pieces of numerical data, since the data points can be counted. Categorical data, on the other hand, are pieces of information collected about a population that can be sorted into different categories or groups. For instance, data collected from students about the name of their cell phone provider or which zip code they live in is categorical data. For each type of data, decide whether the data are numerical or categorical. 1. Number of siblings 2. Hair color 3. Favorite subject 4. Height LESSON 1: CORE ACTIVITY 1. Consider the applicants at Paolo s Pizza Palace. a. How many people applied for jobs at Paolo s Pizza Palace? How many applicants were male? How many were female? b. Does it make sense that most of Paolo s employees are female? 2. There are two options for the gender category (male/female) and two options for job status (hired/not hired). Build a two- way table that models these options.

2 Analyzing bivariate data 3. Are the data in the two- way table categorical or numerical? How do you know? 4. The two- way table displays information about Paolo s Pizza Palace s applicants. Determine the marginal totals and the grand total. Male Female Row totals Not hired 28 28 Hired 10 30 Column totals LESSON 1: CONSOLIDATION ACTIVITY Researchers did a survey of patients at two hospitals, City General and St. John s. The patients were tested to find out if they were infected with a drug resistant bacteria. Of the 185 patients surveyed at City General, 35 were infected with the bacteria. Of the 303 patients surveyed at St. John s, 200 were not infected. Create a two- way frequency table showing the results of the survey and the marginal totals.

Analyzing bivariate data 3 LESSON 1: HOMEWORK Notes or additional instructions based on whole- class discussion of homework assignment: 1. Swimmers and cyclists were surveyed about their participation in other activities. This table shows how many swimmers and cyclists surveyed also participate in another activity. Use the table to answer the following questions. Other activity No other activity Swimmers 58 10 Cyclists 66 18 a. How many swimmers were surveyed? b. How many athletes participate in another activity? c. How many athletes do not participate in another activity? d. How many total athletes were surveyed? e. How many swimmers also participate in another activity? f. How many cyclists do not participate in another activity? g. What proportion of all the swimmers and cyclists also participate in another activity? (Hint: Use your answers from parts b and d.)

4 Analyzing bivariate data 2. This table shows how many voters voted to support an ordinance that bans smoking inside restaurants and on patios. Use the table to answer the following questions. Voted for Voted against Total Voters 60 and older 122 328 Voters 18-59 414 73 Total a. Compute the marginal totals and the grand total. b. How many voters were surveyed altogether? c. How many voters voted for the ordinance? d. How many voters voted against the ordinance? e. How many voters aged 18-59 voted for the ordinance? f. How many voters aged 18-59 voted against the ordinance? g. Do you notice any relationships between the variables? Explain.

Analyzing bivariate data 5 LESSON 1: STAYING SHARP 1. Write the equation of the line shown here. 2. An equation of a line is given in standard form. Write the equation of the line in slope intercept form. Practicing skills & concepts 2x 3y = 6 3. Complete the table of values corresponding to the equation y = 3x 2. 4. A package of bubble gum costs $0.27. Complete the table to show the cost, y, of buying x packages of bubblegum. Preparing for upcoming lessons x - 2 0 4 y Number of packages (x) 3 5 11 Cost (y) 5. Find the slope of the line containing the two points (3,- 2) and (- 5,2). 6. If Jarrod paid $0.96 for 3 apples, what was the price per apple? Reviewing ideas from earlier grades

6 Analyzing bivariate data

Analyzing bivariate data 7 Lesson 2: Analyzing two-way tables LESSON 2: OPENER A school administrator wanted to know what students thought about having a Pajama Friday during homecoming week. She decided to survey a group of freshmen and sophomores. Of the freshmen, 10 said they were in favor of the idea and 15 said they were not in favor of the idea. Of the sophomores, 20 said they were in favor and 5 said they were not in favor. 1. Create a two- way table to represent these data. 2. Find the row and column totals. 3. What percentage of the freshmen said they were in favor of Pajama Friday? 4. What percentage of the sophomores said they were in favor of Pajama Friday? LESSON 2: CORE ACTIVITY 1. Compute the joint relative frequencies for each cell in the table. A hint for getting started is shown in the first cell. Joint Relative Frequencies Male Female Not hired 28/96 = Hired 2. Write sentences to indicate the meaning of each number you calculated in the previous question.

8 Analyzing bivariate data 3. How do you compute marginal relative frequencies? 4. Fill in the marginal relative frequencies in the table. Male Female Row totals Not hired 28 28 56 Hired 10 30 40 Column totals 38 58 96 Row marginal relative frequencies Column marginal relative frequencies 5. Complete the statements to interpret each of the marginal frequencies you found in the previous question. 60 female 58 male 40 42 not hired all hired 40% of applicants were. % of all applicants were female. 42% of applicants were. % of all applicants were not hired. 6. What do these relative frequencies tell you about discriminatory hiring practices at Paolo s Pizza Palace?

Analyzing bivariate data 9 7. Complete the journal entry to record the definitions of joint relative frequency and marginal relative frequency. Description How to find Example Joint relative frequency Marginal relative frequency LESSON 2: CONSOLIDATION ACTIVITY Survey the class and record the gender of each classmate and whether he/she owns a dog or not. 1. Complete the two- way table to model the data you collected. Compute the marginal totals and the grand total. Owns a dog Does not own a dog Row totals Male Female Column totals 2. Use joint relative frequencies to answer the following questions. a. What percentage of students in your class do not own a dog? b. What percentage of students in your class are males who own a dog?

10 Analyzing bivariate data 3. Use marginal relative frequencies to answer the following questions. a. What percentage of those you surveyed are males? b. What percentage of those you surveyed are females? c. What percentage of those you surveyed own dogs? d. What percentage of those you surveyed do not own dogs? 4. Can you determine from the percentages that you calculated whether or not male classmates are more likely to own a dog than female classmates? Justify your response.

Analyzing bivariate data 11 LESSON 2: HOMEWORK Notes or additional instructions based on whole- class discussion of homework assignment: 1. This table shows how many swimmers and cyclists surveyed also participate in another activity. Find the row, column, and grand totals. Other activity No other activity Total Swimmers 58 10 Cyclists 66 18 Total 2. Compute the joint relative frequencies for each cell in the table. Joint Relative Frequencies Other activity No other activity Swimmers Cyclists 3. Write a sentence to indicate the meaning of each number you found in the previous question. 4. Complete the table to calculate the marginal relative frequencies. Other activity No other activity Row totals Row marginal relative frequencies Swimmers 58 10 Cyclists 66 18 Column totals Column marginal relative frequencies 5. Write a sentence to indicate the meaning of each number you found in the previous question.

12 Analyzing bivariate data LESSON 2: STAYING SHARP 1. Write the equation of the line shown here. 2. Write the equation of the line that has a slope of - 3 and a y- intercept of 5. Practicing skills & concepts 3. Complete the table of values corresponding to the equation y = 0.5x - 4. 4. Use the table to predict the cost to buy 10 gallons of gas. Preparing for upcoming lessons x 0 1 2 y Number of gallons of gas (x) Cost (y) 5 $8.80 6 $10.56 7 $12.23 Reviewing ideas from earlier grades 5. Jason was graphing a line. He started at the point (- 4,- 3). Then he used the slope of the line, m = 2/3, to graph additional points. What is one other possible point he could have used to graph the line? 6. On a recent shopping spree, Tyrese bought 4 pairs of shorts and a new pair of sneakers. Each pair of shorts was the same price. He spent a total of $135. If the sneakers cost $55, how much did each pair of shorts cost?

Analyzing bivariate data 13 Lesson 3: Inferences with bivariate categorical data LESSON 3: OPENER On a recent day at the Corner Coffee Shop, Marcus noticed that 52 people were drinking coffee: 13 men and 39 women. Use this information to answer the following questions. Marcus asks the following questions. What are the answers to his questions? If you do not have enough information to answer a question, explain. 1. What percentage of the coffee drinkers were men? 2. What percentage of the coffee drinkers were not men? 3. What percentage of the men were drinking coffee? LESSON 3: CORE ACTIVITY 1. Compute the following conditional relative frequencies: a. The proportion of males that were hired b. The proportion of females that were hired 2. How do you compute conditional relative frequencies? Row: Conditional relative frequency = Column: Conditional relative frequency =

14 Analyzing bivariate data 3. The information in the applicant files for Harry s Hamburger Shack is shown in the table. Find the column and row totals and compute the conditional relative frequencies for being hired based on gender. Do you believe discrimination based on gender is taking place? Explain. Male Female Row totals Not hired 11 28 Hired 10 30 Column Totals LESSON 3: ONLINE ASSESSMENT Today you will take an online assessment.

Analyzing bivariate data 15 LESSON 3: HOMEWORK Notes or additional instructions based on whole- class discussion of homework assignment: Malcolm is doing an independent study project for his algebra class. He is trying to use college basketball statistics from players who made it into the NBA to help predict future draft picks. For one part of his project he wants to analyze the free throw and three- point shooting ability of some players. He takes shooting percentages and categorizes them into three categories: poor, average, and good. The table shows the results of his tallying. Free Throw Shooting Ability Three- point shooting ability Poor Average Good Poor 12 30 12 Average 27 46 47 Good 32 24 6 1. Fill in the column and row totals. 2. How many basketball players did he include in his data? 3. What does the number 46 represent in the context of Malcolm s data collection? 4. What percentage of players were ranked as having good three- point ability and average free throw ability? 5. What percentage of the players were ranked as having good free throw ability? 12 6. What does the proportion = 0.17 represent in this problem?! 71

16 Analyzing bivariate data 7. Refer to the two- way table representing hiring practices at Harry s Hamburger Shack from the Core activity. 11 a. What does the proportion = 0.28 mean in the context of the problem?! 39 40 b. What does the proportion = 0.51 mean in the context of the problem?! 79 30 c. What does the proportion = 0.38 mean in the context of the problem?! 79 8. Complete the journal entry to define conditional relative frequency. Description How to find Example(s) Conditional relative frequency

Analyzing bivariate data 17 LESSON 3: STAYING SHARP 1. Write the equation of the line shown here. 2. Write the equation of the line, in slope- intercept form, that has a slope of - 2 and goes through the point (1,1). Practicing skills & concepts Preparing for upcoming lessons 3. Complete the table corresponding to the equation y = 2x. x - 2 y 0 8 4. Robert created this table to keep track of how many hours he spends mowing lawns. Use the table to determine how many lawns he mowed if he spent 7! hours mowing. Number of lawns mowed (x) Hours spent mowing (y) 2 3 3 4 1 4 6 2! Reviewing ideas from earlier grades 5. Daisy was computing the slope between the two points (1,3) and (4,7). She came up with a slope of 3. Is she correct? If not, what did she do wrong?! 4 6. Maria adopted a puppy from an animal shelter. When she got him, the puppy weighed 3 pounds, but he grew at a rate of 1.5 pounds per week. How much did the puppy weigh after 5 weeks? After 6 weeks?

18 Analyzing bivariate data

Analyzing bivariate data 19 Lesson 4: Exploring residuals LESSON 4: OPENER In a previous lesson, you created a model to represent the relationship between height and shoe size for the students in Tommy s class. You also found the line of best fit. The scatterplot and line of best fit are shown in the graph. 1. What is the predicted shoe size for a student who is 62 tall? How does the predicted value compare to the actual value? 2. What is the predicted shoe size for a student who is 74 tall? How does the predicted value compare to the actual value?

20 Analyzing bivariate data LESSON 4: CORE ACTIVITY 1. Take a closer look at the regression equation you found for the shoe size vs. height data. Use this equation to determine the predicted show size for each student in Tommy s class. Then find the residual (the overestimate or underestimate) for each point. Height in inches Actual Shoe Size Predicted Shoe Size Residual = Actual - Predicted 62 6 0.39(62)- 16.56 = 7.6 6 7.6 = - 1.6 74 13 70 9 67 11 53 4 58 7 2. Complete the residual plot for these data by plotting the residual value for each corresponding student height.

Analyzing bivariate data 21 3. The hours of sunlight Anchorage, Alaska receives over the year are shown in the table. Plot the points and use a graphing calculator to find the line of best fit. Then create a graph of the residual values. What do you notice? Hours of daylight in Anchorage, Alaska Month Hours of daylight Month Hours of daylight January 5.3 July 18.8 February 7.5 August 16.8 March 10.1 September 14 April 13.2 October 11.2 May 16 November 8.2 June 18.3 December 5.7

22 Analyzing bivariate data 4. What does each residual plot tell you about the fit of the regression line to the data? 5. What does the residual plot you created for the shoe size vs. height data tell you about the fit of the regression line to the data?

Analyzing bivariate data 23 LESSON 4: REVIEW ONLINE ASSESSMENT You will work with your class to review the online assessment questions. Problems we did well on: Skills and/or concepts that are addressed in these problems: Problems we did not do well on: Skills and/or concepts that are addressed in these problems: Addressing areas of incomplete understanding Use this page and notebook paper to take notes and re- work particular online assessment problems that your class identifies. Problem # Skills and/or concepts that are addressed in these problems: Problem # Skills and/or concepts that are addressed in these problems: Problem # Work for problem:

24 Analyzing bivariate data LESSON 4: HOMEWORK Notes or additional instructions based on whole- class discussion of homework assignment: Next class period, you will take an end- of- unit assessment. One good study skill to prepare for tests is to review the important skills and ideas you have learned. Use this list to help you review these skills and concepts, especially by reviewing related course materials. Important skills and ideas you have learned in the unit Statistical modeling: Select and use appropriate graphical representations for presenting numerical data Use measures of central tendency, including median and mean, to describe a set of data Use various measures of variability such as range, interquartile range, mean absolute deviation, and standard deviation to describe sets of data Distinguish between and interpret marginal, joint, and conditional relative frequencies Use frequency tables and frequencies to compare different bivariate data sets Prepare and analyze a residual plot to assess the appropriateness of a linear model Homework Assignment Part I: Study for the end- of- unit assessment by reviewing the key ideas listed above. Part II: Complete the online More practice activity in the topic Analyzing bivariate data. Note the skills and ideas for which you need more review, and refer back to related activities and animations from this topic to help you study. Part III: Complete Lesson 4: Staying Sharp.

Analyzing bivariate data 25 LESSON 4: STAYING SHARP 1. Write the equation of the line shown. 2. Write the equation of the line that passes through the points (- 3,1) and (3,5). Practicing skills & concepts 3. Complete the table corresponding to the equation x + 2y = 3. 4. Mandy s Mobile Phone Service charges customers a flat fee of $25 per month plus $0.05 per minute. If a customer was billed $30.00, how many minutes did he talk? Preparing for upcoming lessons x 0 1 y 0 Reviewing ideas from earlier grades Austin is building a fence around his yard. He wants to figure out how much this project will cost. He created the following table to show the total cost, y, to buy x number of fence boards. Number of fence boards (x) Cost (y) 10 $21.50 20 $43.00 30 $64.50 Use this table to answer questions 5 and 6. 5. Find the rate of change (slope) of the data. 6. What does the rate of change (slope) represent in the context of this problem?

26 Analyzing bivariate data

Analyzing bivariate data 27 Lesson 5: Checking for understanding LESSON 5: OPENER Match each residual plot with the best description. END-OF-UNIT ASSESSMENT Today you will take an end- of- unit assessment.

28 Analyzing bivariate data LESSON 5: CONSOLIDATION ACTIVITY Many doctors are concerned that students develop back and shoulder problems when they carry backpacks that are too heavy. Some doctors recommend that the weight of a student's backpack should not exceed 10% of the student's body weight. The Health Sciences Club at Smithville High School decides to check the weights of its members' backpacks. They collect data and find that the line of best fit is Backpack weight = 0.091(Student s weight) + 16.265. 1. Use this equation to determine the predicted backpack weight for each student. Then find the residual (the overestimate or underestimate) for each point. Student s Weight (lbs.) Backpack Weight (lbs.) 120 26 187 30 109 26 103 24 131 29 165 35 158 31 116 28 Predicted Backpack Weight Residual = Actual - Predicted 2. Complete the residual plot for these data by plotting the residual value for each corresponding student weight. 3. What does the residual plot you created tell you about the fit of the regression line to the data?

Analyzing bivariate data 29 LESSON 5: HOMEWORK Notes or additional instructions based on whole- class discussion of homework assignment: Sanborn s long- nosed bats are found in Arizona. These bats feed on the nectar of flowers. In a study of the feeding habits of these bats, scientists recorded the weights of nine bats before gathering nectar. After five minutes of feeding, the scientists weighed the bats again to determine the amount of nectar consumed. The table shows the data collected by the scientists. The scientists find that the line of best fit is Weight gain = 0.2(Pre- feeding weight) - 2.17. 1. Use this equation to determine the predicted weight gain for each recorded pre- feeding weight. Then find the residual (the overestimate or underestimate) for each. Pre- feeding weight (p) Weight gain (g) Predicted weight gain Residual 18.2 2.0 20.2 2.1 19.0 1.5 17.3 1.2 17.8 1.2 19.5 2.1 21.0 1.8 17.0 1.4 18.3 1.0 2. Complete the residual plot for these data by plotting the residual value for each corresponding pre- feeding weight. 3. What does the residual plot you created tell you about the fit of the regression line to the data?

30 Analyzing bivariate data LESSON 5: STAYING SHARP Practicing skills & concepts 1. Write the equation of the line shown. 2. Write the equation of the line, in slope- intercept form, that is perpendicular to the line y = - 3x + 1 and passes through the point (- 3,- 2). Preparing for upcoming lessons 3. Haley goes to the store to buy snacks for her birthday party. She wants to buy several bags of chips and a large container of dip. Each bag of chips costs $1.50 and the dip costs $5.75. Complete the table to show how much she will spend (y) if she buys x bags of chips and a container of dip. Number of bags of chips (x) 1 3 5 How much she will spend (y) 4. Write an equation to represent how much Haley will spend (y) if she buys x bags of chips and a container of dip. 5. Use the table you constructed in question 3 to figure out how many bags of chips Haley purchased if she spent a total of $11.75. Reviewing ideas from earlier grades 6. When Allen was 10 years old, he was 55 inches tall. When he was 14 years old, he was 63 inches tall. How many inches did he grow per year between the ages of 10 and 14?