Relating Quantitative Data

The Lesson Activities will help you meet these educational goals:

  • Mathematical Practices—You will make sense of problems and solve them, reason abstractly and quantitatively,and use mathematics to model real-world situations.
  • Inquiry—You will perform an investigation in which you will analyze data;communicate results in tables, graphs, and written form; and draw conclusions.
  • STEM—You will apply mathematical and technology tools and knowledge to analyze real-world situations and gain insight into careers in science, technology, engineering, and math.
  • 21st Century Skills—You will employ online tools for research and analysis,carry out technology-assisted modeling, use critical-thinking and problem-solving skills, and communicate effectively.

Directions

You will evaluate some of these activities yourself, and your teacher may evaluate others. Please save this document before beginning the lesson and keep the document open for reference during the lesson. Type your answers directly in this document for all activities.

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Self-Checked Activities

Read the instructions for the following activities and type in your responses. At the end of the lesson, click the link to open the Student Answer Sheet. Use the answers or sample responses to evaluate your own work.

  1. Plotting and Fitting Data

In this activity, you will create a scatterplotof the data in the table below.First you will look for patterns in the numbers and figure out what the mathematical relationship might be. Then you will use this relationship to create a mathematical modelto help manage the bear population.

Year / LicensesIssued / BearsHarvested
1991 / 7,757 / 2,143
1994 / 9,826 / 2,329
1997 / 11,440 / 3,212
2001 / 16,510 / 4,936
2004 / 13,669 / 3,391
2007 / 11,936 / 3,172
2010 / 9,689 / 2,699
  1. You have two variables of interest: the number of hunting licenses issued and the number of bears harvested. Which one should be the independent variable and which should be the dependent variable? Why?

Type your response here:

  1. Is there a reason for not including the year as a variable of interest?

Type your response here:

  1. Using the Scatter Plottool, plot the number of licenses issued along the x-axis and the number of bears harvested along the y-axis. If you need help, follow these instructionsfor using the online probability tools.

Be sure to title your graph and label the axes. Export your finished graph to a file, and then paste it in the space below.

Type your response here:

  1. What type of relationship seems to exist between the licenses purchased and the number of bears harvested in these years?

Type your response here:

  1. Is there a linear relationship between the two variables? Why?

Type your response here:

  1. Assuming a linear relationship between licenses issued and bears harvested, write an equation that relates thenumber of bears harvested to the number of licenses issued. Assume that when zero licenses are issued, zero bears are killed. In that case, what is the value of the y-intercept (the value of b)?

Type your response here:

  1. With a simplified equation of y = mx, the slope of the line from (0,0) to each of these points is (rise over run).Each of these points would result in a somewhat different slope, though. Calculate theslopes and record them in the table.

Type your response here:

Year / Value of m
1991
1994
1997
2001
2004
2007
2010
  1. As you saw in the knowledge article, finding the line of best fit for a set of data is a fairly sophisticated process. But you can often get a fairly reasonable answer by taking the arithmetic average (mean) of all the slopes. Calculate the mean of the slopes and use it to give the equation of the line of best fit.

Type your response here:

Note: If possible, keep this graph available on your computer for the next part of the Lesson Activities. That way, you won’t have to plot this graph again.

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

  1. Visually Assessing Fit
  2. If you still have the graph from the first part of the Lesson Activity available, open it. Otherwise, recreate the graph:Use the Scatter Plottool and the data from the table in activity 1.Plot the number of licenses issued along the x-axis and the number of bears harvested along the y-axis. Label the graph and the axes appropriately.

Next, check the box for Custom Fit and enter this expression forthe linear equation:

y = 0.27x(remember to use * for multiplying with these tools).

Export your finished graph to a file, and paste it in the space below.

Does this line seem to fit the data well?

Type your response here:

  1. Now try the best fit line calculated by the Scatter Plot tool. Check the box for Line of Best Fit to display it along with your custom equation line. Export your finished graph to a file, and paste it in the space below.

Visually, which line seems to fit your data better? Explain your reasoning.

Type your response here:

Hint: For the next two questions, you may wish to copy and paste the data from the tables below into a spreadsheet. Then you can copy and paste your results back into this document.If you know how to use it, a spreadsheet can speed up work like this because it is designed to operate on large sets of data. A spreadsheet is a critical tool for data analysis like this.

  1. Now, to check your visual assessment, look at residuals for your custom equation using the table below. (The first row is filled in for you.) Specifically, calculate and enter values for the three incomplete columns:
  • Calculate estimated bear harvest (yest) using your equation: yest= 0.27x.
  • Calculate the residual for each data point: ydata–yest.
  • Calculate the absolute value for your residual (change any negatives to positives).
  • Finally, to informally evaluate these residuals, total the values in the last two columns. The smaller these sums, the closer the fit.

Type your response here:

Year / Licenses (x) / Bears
(ydata) / Est. Bears
(yest) / Residual
(ydata–yest) / Abs. Value
of Residual
1991 / 7,757 / 2,143 / 2,094 / 49 / 49
1994 / 9,826 / 2,329
1997 / 11,440 / 3,212
2001 / 16,510 / 4,936
2004 / 13,669 / 3,391
2007 / 11,936 / 3,172
2010 / 9,689 / 2,699
Sum
  1. Now analyze the residuals for the best fit equationy = 0.308x– 435 to complete the table.

Type your response here:

Year / Licenses (x) / Bears
(ydata) / Est. Bears
(yest) / Residual
(ydata – yest) / Abs. Value
of Residual
1991 / 7,757 / 2,143 / 1,954 / 189 / 189
1994 / 9,826 / 2,329
1997 / 11,440 / 3,212
2001 / 16,510 / 4,936
2004 / 13,669 / 3,391
2007 / 11,936 / 3,172
2010 / 9,689 / 2,699
Sum
  1. Neither sum is enough by itself to determine how well a line fits data. The sum of the residuals (column 5) shows how evenlythe data issplit above and below the line. The sum of the absolute values of the residuals shows the total distance from the line for the entire set of points. Both are useful to determine fit.

From what you now know, discuss how your custom line fits compared with the computer-calculated best fit line.

Type your response here:

You’re confidentnow that the best-fit linear equation fits the data closely,so you can use it as a mathematical model for planning and for managing the bear population. For the following problems, use this best-fit relationship:y = 0.308x – 435, where y is the number of bears harvested and x is the number of licenses issued.

  1. If the bear population estimates are down and the state wants to reduce the harvest to about 1,000 bears, how many licenses should the state issue?

Type your response here:

  1. One year, the weather is especially good, and food for bears is plentiful. The Department of Natural Resources estimates an explosion in the bear population and issues 8,000 licenses.How many bears do you estimate will be harvested?

Type your response here:

Finally, if you’d like to try some quick visual guessing, go to this regressionactivity.Use your cursor to draw a straight line for the scatter plot data, and then select Draw regression line. See how close you came to the best-fit line.

How did you do? Check a box below.

Nailed It!—I included all of the same ideas as the model response on the Student Answer Sheet.

Halfway There—I included most of the ideas in the model response on the Student Answer Sheet.

Not Great—I did not include any of the ideas in the model response on the Student Answer Sheet.

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