Do Now 18 Do Now 18 Balance Point Directions: Use the data table to answer the questions. 1. Calculate the balance point.. Explain whether it is reasonable to fit a line to the data.. The data is plotted as a scatterplot below. Draw a line that contains your balance point and seems to be a good approximation of the data.. Find the equation of the line you plotted in using point- slope form of a line. 1 1 8 6 1 Math Week 7 Packet Page 8
Linear Regression on the Calculator Classwork and Homework We have been finding lines of best fit for quantitative data. Now, we will use our calculators to make the process easier. Math Week 7 Packet Page 9
1. This table shows the average number of gallons of milk a family drinks per week. a. Follow directions on first page to find the equation of the best-fit line. Write down the equation. family size number of gallons of milk 1 1. 1...8.7 6. b. What is the y-intercept of this line? Explain what it represents about the situation. Does it make sense? c. What is the slope of this line? Explain what it represents about the situation. Does it make sense? d. Sally has eight people in her family. How many gallons of milk do you think they drink in a week? Does your answer make sense? If Sally says her family drank 8 gallons of milk last week, do you believe her? e. You run into your friend John at the store and he is buying gallons of milk for his family this week. How many people do you think are in his family? Does your answer make sense? If John says his family has people in it, do you believe him? Math Week 7 Packet Page 1
. Use your calculator to find the best-fit line equation for the data in the table. a. Write down the equation. x y 1 1 1 b. Graph the data points and the best-fit line together on your calculator screen. (If you don t see five points and a line, try pressing [ZOOM][9] to fix.) Draw your calculator screen below.. The table shows ages and heights for several maple trees. Let x = tree age (in years), y = tree height (in feet). a. Enter the data table into your calculator, and find the best-fit line equation: b. Use the equation to predict the height of a 6- year-old tree. Does this make sense? tree age tree height 6 7 6 1 18 9 17 1 11 87 7 c. Use the equation to estimate the age of a tree that is 1 feet tall. Does this make sense? Math Week 7 Packet Page 11
. This table shows the amount spent on TV advertising in the USA in various years. Year 199 1997 1999 1 TV advertising expenditures (in millions of dollars),7,9 6,8,1 a. Enter the data table into your calculator, and find the best-fit line equation: b. Use the equation to predict the amount spent on advertising in 1996. Does this make sense? c. Use the equation to predict in what year the money spent on advertising will reach 6, million dollars. Does this make sense?. The data table shows the number of farmers and the number of cows at each of ten different farms. This problem is about how these two numbers are related. Let x = the number of farmers, y = the number of cows. a. Find the best-fit line equation. You may round off the numbers to three decimal places. b. Predict how many cows there would be at a farm with 6 farmers. Show work. number of farmers number of cows 1 8 8 1 7 7 1 1 8 19 17 11 1 8 c. Predict how many farmers there would be at a farm with 1 cows. Show work. Math Week 7 Packet Page 1
Correlation: Examining Residuals and Correlation Coefficient Today, we will be more closely examining best-fit lines. Begin by creating a scatterplot for the data given in the Excel file on the web site. Remember to label your axes appropriately. Lately, we have been using our calculators to find equations of lines of best fit. Excel can calculate best-fit lines (trendlines) as well. Now, we are going to examine part of the procedure that the calculator and the computer use to find them. Excel can make this process easier for us. Residuals are the: Car Crashes y =.91x - 181.1 R² =.7671 # Deaths (per every 1, people) Year Math Week 7 Packet Page 1
The goal of finding the best-fit line is to make the numbers in the residual column as small as possible. Since some sets of data can contain hundreds of points, you need a single number that represents the error for the entire data set. Adding the individual residuals does not really help. One prediction that is 1, too high offsets a prediction that is 1, too low. A line of best fit is the graph of the linear equation that shows the relationship between two sets of data most accurately (with the least errors). When working with a line of best fit we want to assess how well the line is fitting the data. The sum of square residuals (and thus positive numbers) is one way to measure, but the minimum value that this can reach changes depending on the data. Instead, we use what is called the linear correlation coefficient (r), which is a number that is always between 1 and 1. Note: When calculating r, the linear correlation coefficient, the calculator uses the sum of square residuals. They are related, but we are not going to go into how. The number r, called the correlation coefficient, is a measurement of how closely the bestfit line fits the data. A value of r = 1 or 1 would say the line fits the data perfectly (perfect correlation). A value of r that is greater than.8 is considered strong correlation A value of r that is less than. is often considered weak correlation A value of r = would stand for the worst possible fit (no correlation) o A positive value of r would mean: o A negative value of r would mean: r = 1 r =.99 r =.9 r =. 9 r =.7 r =. Our calculator will tell you r for any best-fit line. To turn this feature on: Math Week 7 Packet Page 1
Enter your original Excel data into your calculator and find the line of best fit. You will now notice that the calculator displays r along with the equation. Equation: What value of r does your calculator display? What does this number tell you? The calculator also display r, the coefficient of determination. This allows you to determine how certain one can be in making predictions from the best-fit line. This is the number that Excel gives you. You can find r by taking the square root of r. Now you try it with some data about the weather and the number of visitors to a water park: What is the best-fit line that Excel gives you? Visitors to a Water Park What is the R value that Excel gives you? Temperature (F) # of Visitors (thousands) Predicted Residuals 6 6.8 What is the value of r? 67.9 69 6.9 7 7.1 What would you say about the correlation of the best-fit line to the data? 7. 77. 79.8 8. 8 8. 8 1. 88 7.7 9 7. 96 6.1 98 8.9 Math Week 7 Packet Page 1
Homework 1. The table of data comes from an experiment by Tor Carlson in 191 on Saccharomyces cerevisiae (a type of yeast). The data show the number of hours elapsed and the number of yeast cells per square unit of area in a Petri dish. a. Find the balance point of the data and add it to the table. b. Make a scatterplot of this data below. Hours Yeast Density 9.6 1 18. 9. 7. 71.1 119.1 6 17.6 7 7. c. Using your calculator, find the line of best fit. Write the equation and r-value below, and plot the best-fit line on your scatterplot. d. With a different color pen or highlighter, mark the residuals on your scatterplot. e. Either in Excel, with your calculator or by hand calculate all of the residuals and label them in your scatterplot. You can either fill out the table below, or print out your Excel work and paste it below. Hours Yeast Density Predicted Residuals Density 9.6 1 18. 9. 7. 71.1 119.1 6 17.6 7 7. Math Week 7 Packet Page 16
f. Look at your r-value. What does this tell you about your best-fit line? (Make sure you address the sign of your r-value and the magnitude of your r-value.). This table gives the mean height in centimeters of boys ages to 1 in the United States. (Source: National Center for Health Statistics) a. Using your calculator find the best-fit line. a. What does the slope represent? Does this make sense? b. What does the y-intercept represent? Does this make sense? Age Height (cm) 19. 6 11.7 7 1. 8 18.1 9 1.7 1 18.8 11 1.7 1 19. 1 16. c. What does the data point (, 19.) represent? Does this make sense? d. Predict the height of a 1-year old. Does this make sense? e. Predict the height of a -year old. Does this make sense? f. Predict how old a 1cm tall boy is. g. What is the correlation coefficient, r, for your best-fit line? h. What does r tell you about your best-fit line? Math Week 7 Packet Page 17
. Explain how your calculator finds the line of best-fit using residuals.. For the scatterplots below estimate the r-value for the lines of best fit. d. e. g. h. i. Math Week 7 Packet Page 18
Two-variable Data Test Review You should be able to: Distinguish between categorical and quantitative data Construct two-way tables for categorical data Calculate joint frequencies, marginal distributions, and conditional distributions Construct segmented bar graphs Determine whether two categorical variables are associated Construct scatterplots for quantitative data Find lines of best fit for data that is approximately linear Determine the equation of a line in point-slope and slope-intercept forms o Point-slope: y = m(x x 1 ) + y 1 o Slope-intercept: y = mx + b Determine whether two quantitative variables are correlated Determine and interpret the correlation coefficient Explain the difference between correlation and causation Make appropriate predictions based on best-fit lines Answer practical questions about data and justify your reasoning 1. The following is data collected on time spent on homework (min) and time spent watching TV (min) each night. Homework (min) Television (min) 7 8 1 1 11 7 6 6 9 7 8 6 6 8 1 9 8 7 8 8 9 1 7 1 6 a. Construct a scatterplot for the data. USE GRAPH PAPER! b. Explain your choices for which variable is on which axis and the scales of each axis. c. Draw in an appropriate line of best fit on your scatterplot. d. Write the equation of the best-fit line in point-slope form. e. Change into slope-intercept form. Math Week 7 Packet Page 19
f. Find the line of best fit given by your calculator. How close is your line from part e to the one given by the calculator? g. What is the slope of the line from your calculator? Write a sentence explaining what it represents about homework and watching television. Does this make sense? h. What is the y-intercept of the line from your calculator? Write a sentence explaining what it represents about homework and watching television. Does this make sense? i. Use your calculator to find the correlation coefficient for this data. Explain what it tells you about the data. Math Week 7 Packet Page
. The following chart shows data from a telephone survey of 16 American adults. Respondent Gender Party Affiliation Person #1 Female Democrat Person # Female Democrat Person # Female Democrat Person # Female Republican Person # Male Republican Person #6 Male Democrat Person #7 Female Democrat Person #8 Female Republican Person #9 Female Republican Person #1 Male Democrat Person #11 Male Democrat Person #1 Male Republican Person #1 Female Republican Person #1 Male Democrat Person #1 Male Republican Person #16 Male Republican a. Construct a two-way table for this data. b. Choose one of the joint frequencies in your table. Write a complete sentence explaining what it represents about this survey. c. Choose one of the rows or columns and calculate the conditional distribution. Write a complete sentence explaining what each value calculated represents about this survey. d. Determine whether gender and party affiliation are associated or independent. Show your work by constructing segmented bar graphs. Explain your reasoning. Math Week 7 Packet Page 1
. The following chart shows nutrition information for items at fast food restaurants. a. Using your calculator, generate a scatterplot comparing total calories and carbohydrates. Estimate the correlation coefficient for theses variables and explain your reasoning. b. Determine the equation for the line of best fit and the correlation coefficient from your calculator. Write them below. How closely did your estimate match the actual correlation coefficient? c. A Bacon Cheeseburger from Five Guys has 9 calories. Use your line of best fit to predict the grams of carbohydrates in this burger. Is this an appropriate prediction? Math Week 7 Packet Page
d. An Original Slider from White Castle has 1 grams of carbohydrates. Use your line of best fit to predict the total calories in this burger. Is this an appropriate prediction?. A survey was conducted about whether a person was an organ donor and whether they were married or single. The two-way table below summarize the data collected. a. Fill in the missing boxes in the table. b. What percentage of single people are not organ donors? Show or explain how you got your answer. c. What percentage of organ donors are married? Show or explain how you got your answer. d. What percentage of people surveyed are organ donors? Show or explain how you got your answer. e. Determine whether martial status and organ donor status are associated or independent. Show your work by constructing segmented bar graphs. Explain your reasoning.. Play the calculator game from class again. Did your performance improve? J Math Week 7 Packet Page