March 14 th March 18 th Unit 8: Linear Functions
Jump Start Using your own words, what is the question asking? Explain a strategy you ve learned this year to solve this problem. Solve the problem! 1
Scatter Plots Reminder: A bivariate data set consists of observations on two variables. We can use a graph called a scatter plot to analyze data and see if a relationship exists between two variable. When we make a scatter plot, we do not connect the data points! Example 1: Make a scatter plot using the table below of price of the shoes and their quality rating and then answer the questions in the box below. Let x represent the price (in dollars) of the shoes and y represent the quality rating of the shoes. The quality rating is on a scale of 0 to 100, with 100 being the highest quality. Shoe Price Quality (dollars) Rating 1 65 71 2 45 70 3 45 62 4 80 59 5 110 58 6 110 57 7 30 56 8 80 52 9 110 51 10 70 51 a) One observation in the data set is 110,57. What does this ordered pair represent in terms of cost and quality? b) Is there a relationship between price and the quality of athletic shoes? 2
Example 2: Data were collected on = shoe size and = score on a reading-ability test for 30 elementary school students. The scatter plot of these data is shown below. a) Does there appear to be a statistical relationship between shoe size and score on the reading test? b) Explain why it is not reasonable to conclude that having big feet causes a high reading score. Can you think of a different explanation for why you might see a pattern like this? Statistical tical Relationship A pattern in a scatter plot indicates that the values of one variable tend to vary in a predictable way as the values of the other variable change. This is called a statistical relationship. In the last example, reading scores increased as shoe size increased. This is useful information, but be careful not to jump to the conclusion that increasing one s shoe size causes a higher reading score. There may be some other explanation for this! Statistical relationships do not mean that one variable causes something to happen to the other. 3
Example 3: Below are examples of the three types of statistical relationships... 4
Independent Practice The table below shows the price and overall quality rating for 15 different brands of bike helmets. Data Source: www.consumerreports.org Helmet Price Quality (dollars) Rating A 35 65 B 20 61 C 30 60 D 40 55 E 50 54 F 23 47 G 30 47 H 18 43 I 40 42 J 28 41 K 20 40 L 25 32 M 30 63 N 30 63 O 40 53 a) One observation in the data set is 30,47. What does this ordered pair represent in terms of price and quality? b) Do you think that there is a statistical relationship between price and quality rating? If so, describe the nature of the relationship. 5
Construct a scatter plot of price ( ) and quality rating ( ). Use the grid below. 6
Jump Start Using your own words, what is the question asking? Explain a strategy you ve learned this year to solve this problem. Solve the problem! 7
Patterns in Scatter Plots Example 1: Take a look at the following five scatter plots. Answer the 3 questions for each scatter plot. Scatter Plot 1 Is there a relationship? If there is a relationship, does it appear to be linear? If the relationship appears to be linear, is it a positive or negative linear relationship? Scatter Plot 2 Is there a relationship? If there is a relationship, does it appear to be linear? If the relationship appears to be linear, is it a positive or negative linear relationship? 8
Scatter Plot 3 Is there a relationship? If there is a relationship, does it appear to be linear? If the relationship appears to be linear, is it a positive or negative linear relationship? Scatter Plot 4 Is there a relationship? If there is a relationship, does it appear to be linear? If the relationship appears to be linear, is it a positive or negative linear relationship? 9
Scatter Plot 5 Is there a relationship? If there is a relationship, does it appear to be linear? If the relationship appears to be linear, is it a positive or negative linear relationship? Example 2: *OUTLIERS: An outlier is an unusual point in a scatter plot that does not seem to fit the general pattern or that is far away from the other points in the scatter plot. The scatter plot below was constructed using data from a study of Rocky Mountain elk ( Estimating Elk Weight from Chest Girth, Wildlife Society Bulletin, 1996). The variables studied were chest girth in centimeter ( ) and weight in kilogram ( ). Do you notice any point in the scatter plot of elk weight versus chest girth that might be described as an outlier? If so, circle it and label it outlier. Explain why you would consider this an outlier. 10
Independent Practice 1) The scatter plot below was constructed using data size in square feet ( ) of several houses and price in dollars ( ). Describe the relationship between price and size for these houses. Are there any noticeable outliers? 2) The scatter plot below was constructed using data on length in inches ( ) of several alligators and weight in pounds ( ). Describe the relationship between weight and length for these alligators. Are there any noticeable outliers? 11
3) The scatter plot below was constructed using data on age in years ( ) of several Honda Civics and price in dollars ( ). Describe the relationship between price and age for these cars. Are there any noticeable clusters or outliers? Tuesday Exit Ticket Which of the following scatter plots shows a positive linear relationship? Explain how you know. Scatter Plot 1 Scatter Plot 2 Scatter Plot 3 Scatter Plot 4 12
Jump Start Using your own words, what is the question asking? Explain a strategy you ve learned this year to solve this problem. Solve the problem! 13
Line of Best Fit A line of best fit is a line that is close to the majority of data points in a scatter plot. Example 1: July Rainfall and Temperatures in Selected Midwestern Cities a) Explain what the point (31, 64) means in this problem. b) The line has been drawn to model the relationship between the amount of rain and the temperature in those Midwestern cities. Use the line to predict the mean July temperature for a Midwestern city that has a mean of 32 inches of rain per year. c) For which of the cities in the sample will the line do the worst job of predicting the mean temperature? The best? Explain your reasoning with as much detail as possible. 14
Example 2: Scientists captured a small sample of alligators and measured both their length (in inches) and weight (in pounds). Torre used their data to create the following scatter plot and drew a line to capture the trend in the data. She and Steve then had a discussion about the way the line fit the data. What do you think they were discussing and why? Alligator Length (in.) and Weight (lb.) 700 650 600 550 Weight (pounds) 500 450 400 350 300 250 200 150 100 50 0 0 50 60 70 80 90 100 110 Length (inches) 120 130 140 150 Independent Practice 1) The local ice cream shop keeps track of how much ice cream they sell versus the noon temperature on that day. a) Is the line drawn a good line of best fit? Explain. b) Using the line of best fit, what would be a good prediction of how much money the ice cream shop will make if the temperature is 16 C? 15
2) The graph below shows a scatterplot of the birth rate of different countries versus the yearly production amount per person (in dollars). Is the line of best fit that is drawn a good line of best fit for the data? Explain. 3) The graph below shows a student s approximation of the line of best fit for the data in the table. Describe how the student could make the line more accurately fit the data. 4) The scatter plot shows the average price of a major league baseball ticket from 1997 to 2006. Using the line of best fit, what would be a good estimate of the price of a ticket in 2003? How does that compare to the actual price of a ticket in 2003? 16
Jump Start Using your own words, what is the question asking? Explain a strategy you ve learned this year to solve this problem. Solve the problem! 17
Unit 8 Review Questions 1) The cost to rent a paddleboat at the city park includes an initial fee of $7.00, plus $3.50 per hour. a) Write an equation that models the relationship between the total cost, y, and the number of hours, x that the paddleboat is rented. b) Describe the rate of change for this problem. (Say what the rate of change is and what it means) 2) The scatter plot below shows the number of customers in a restaurant for four hours of the dinner service on two different Saturday nights. The line shown models this relationship, and x = 0 represents 7 p.m. a) Explain what the y-intercept represents. b) Using the line of best fit, what would be a good prediction of the number of customers dining 1.5 hours after 7 p.m.? 3) 18
4) Which line represents the best fit for the scatter plot data? 5) 19
6) 7) 20
8) (Hint: Translate the statement into an equation and then use the equation to make a table of a few x- values and y-values that you can use to make your graph!) 21
Week 26 Homework At a dinner party, every person shakes hands with every other person present. a. If three people are in a room and everyone shakes hands with everyone else, how many handshakes will there be? b. Make a table for the number of handshakes in the room for one to six people. You may want to make a diagram or list to help you count the number of handshakes. Number People Handshakes 1 2 3 4 5 6 c. Make a scatter plot of number of people ( ) and number of handshakes ( ). d. Does the trend seem to be linear? Why or why not? Are there any outliers? 22
2. A scatter plot is prepared to show the relationship between x and y where x represents a student score on a test and y represents the number of incorrect answers a student received on the same test. Which of the choices is the correct scatter plot for this situation? 23