Algebra 2 – Linear Models Activity – (Total Points 36)
Name ___________________________________ Date ___________
Problem 1: Manatees are large, gentle sea creatures that live along the Florida coast. Many manatees are killed or injured by powerboats. Here are the data on powerboat registrations (in thousands) and the number of manatees killed by boats in Florida in the years 1977 to 1990
Year Powerboat Registrations (1000’s) Manatees Killed
1977 447 13
1978 460 21
1979 481 24
1980 498 16
1981 513 24
1982 512 20
1983 526 15
1984 559 34
1985 585 33
1986 614 33
1987 645 39
1988 675 43
1989 711 50
1990 719 47
A) Enter Data into graphing calculator. (1pt)
B) Make a scatter plot using the graphing calculator, make a sketch below, and describe the relationship. (2pts)
C) Calculate the linear correlation coefficient and interpret the results. (2pts)
D) Calculate the least-squares regression line using the graphing calculator and draw the trend line on the scatter plot and on the sketch above. (2pts)
E) Interpret the Slope. (2pts)
F) Interpret the y-intercept. (1pt)
G) Prediction: If the number of powerboats is 800 thousands, how many manatees are predicted to be killed? (2pts)
Problem 2: Emission rates of carbon monoxide in grams per mile were taken to see whether the emission rate increases with the mileage of a car. The carbon monoxide emission rate(y) and the mileage(x) were recorded for 22 cars of the same make and model.
A) Enter Data into graphing calculator. (1pt)
B) Make a scatter plot (next page) using the graphing calculator, make a sketch below, and describe the relationship. (2pts)
D) Calculate the least-squares regression line using the graphing calculator and draw the trend line on the scatter plot and on the sketch above. (2pts)
E) Interpret the Slope. (2pts)
F) Interpret the y-intercept. (1pt)
G) Prediction: If the number of miles driven is 7500 miles, how much carbon monoxide is emitted? (2pts)
Problem 3: This data set came from a real situation in the food industry. It was from the joint work by two of the authors of an article in a project for designing a fast and inexpensive computer vision oyster volume
estimation system using 3-dimensional (3-D) information. Volume measurement or estimation is an important process in the food industry, since the price of the product is often determined by the size of the product. For instance, the price of oyster meat is determined by its size. Large oyster meat can sell for a higher price than regular or small sizes. Before computer vision sorting systems were available for oyster sorting, human vision was the main measurement tool used for oyster sorting. But manual sorting is prone to errors which result in both profit loss and customer complaints. Parr et al. (1994) implemented the idea of using a 2-dimensional (2-D) digital image of the oyster through a machine vision system to estimate its size. The system takes the digital image of oyster meat that is put on a conveyor belt from a camera directly above the meat looking down. The system binarizes the image taken from the digital camera so that the pixels (i.e., picture elements) of the oyster meat are set to 1 and the pixels of the background are set to 0. Sample digital images before and after
binarization are shown in Figure 1. The 2-D area of the oyster meat can then be measured by counting the number of pixels that have the value 1. This 2-D area measurement of oyster meat in pixels is then correlated with the actual volume of the oyster meat to find an equation for estimating the oyster meat volume. In this study, the independent variable is the actual volume of oyster meat and the dependent variable is the area in pixels (for 2-D) or volume in pixels (for 3-D) reported from the computer vision volume estimation system. We have identified the pixel count to be the dependent variable because its value depends on the actual volume of the oyster meat, the independent variable. The pixel count reported by the 2-D or 3-D volume estimation system is used with the calibration equation to estimate the actual volume of the oyster meat. Visual inspection (with a scatter plot) is usually a preliminary step to fitting a straight line model since in some situations the relationship between the two variables may be non-linear. We can compute the Pearson correlation coefficient, find the equation of the straight line that best fits the data using the actual volume and the volume in pixels for 3D, and compute the coefficient of determination, R2. Journal of Statistics Education Volume 17, Number 2 (2009),
www.amstat.org/publications/jse/v17n2/datasets.chang.html.
Original Image Binarized Image Figure 1. Digital Images for a 2-D system. A) Enter Data (next page) into graphing calculator. (1pt)
B) Make a scatter plot (next page) using the graphing calculator, make a sketch below, and describe the relationship. (2pts)
D) Calculate the least-squares regression line using the graphing calculator and draw the trend line on the scatter plot and on the sketch above. (2pts)
Oyster ID Oyster Weight (g)
Oyster
Volume Pixels 3D Pixels 2D
1 12.92 13.04 5136699 47907
2 11.4 11.71 4795151 41458
3 17.42 17.42 6453115 60891
4 6.79 7.23 2895239 29949
5 9.62 10.03 3672746 41616
6 15.5 15.59 5728880 48070
7 9.66 9.94 3987582 34717
8 7.02 7.53 2678423 27230
9 12.56 12.73 5481545 52712
10 12.49 12.66 5016762 41500
11 10.12 10.53 3942783 31216
12 10.64 10.84 4052638 41852
13 12.99 13.12 5334558 44608
14 8.09 8.48 3527926 35343
15 14.09 14.24 5679636 47481
16 10.73 11.11 4013992 40976
17 15.17 15.35 5565995 65361
18 15.50 15.44 6303198 50910
19 5.22 5.67 1928109 22895
20 7.75 8.26 3450164 34804
21 10.71 10.95 4707532 37156
22 7.91 7.97 3019077 29070
23 6.93 7.34 2768160 24590
24 13.63 13.21 4945743 48082
25 7.67 7.83 3138463 32118
26 11.27 11.38 4410797 45112
27 10.98 11.22 4558251 37020
28 8.87 9.25 3449867 39333
29 13.68 13.75 5609681 51351
E) Interpret the Slope. (2pts)
F) Interpret the y-intercept. (1pt)
G) Prediction: If the actual volume is 16 cubic units, what is the predicted volume calculated by the computer of the 3-D image in pixels? (2pts)
Oyster_ID Oyster_Weight_g Oyster_Volume_cc Pixels_3D Pixels_2D 1, 12.92, 13.04, 5136699 47907 2 11.40 11.71 4795151 41458 3 17.42 17.42 6453115 60891 4 6.79 7.23 2895239 29949 5 9.62 10.03 3672746 41616 6 15.50 15.59 5728880 48070 7 9.66 9.94 3987582 34717 8 7.02 7.53 2678423 27230 9 12.56 12.73 5481545 52712 10 12.49 12.66 5016762 41500 11 10.12 10.53 3942783 31216 12 10.64 10.84 4052638 41852 13 12.99 13.12 5334558 44608 14 8.09 8.48 3527926 35343 15 14.09 14.24 5679636 47481 16 10.73 11.11 4013992 40976 17 15.17 15.35 5565995 65361 18 15.50 15.44 6303198 50910 19 5.22 5.67 1928109 22895 20 7.75 8.26 3450164 34804 21 10.71 10.95 4707532 37156 22 7.91 7.97 3019077 29070 23 6.93 7.34 2768160 24590 24 13.63 13.21 4945743 48082 25 7.67 7.83 3138463 32118 26 11.27 11.38 4410797 45112 27 10.98 11.22 4558251 37020 28 8.87 9.25 3449867 39333 29 13.68 13.75 5609681 51351 30 14.27 14.37 5292105 53281
