HCPSS Worthwhile Math Task
AP Statistics, Unit 3 Essential Curriculum Objectives: The student will be able to

Construct a scatterplot to display the relation between two quantitative variables, including adding a categorical variable.

Interpret the direction, form, and strength of the association between variables.

Define and use appropriate notation for a regression line and the least-squares regression line.

Calculate the least-squares regression line, using a calculator, and interpret the slope and intercept.

Appropriately use the regression line to predict y for a given x .

Calculate and plot residuals to identify patterns of fit.
MP1: Make sense of problems and persevere in solving them.
MP2: Reason abstractly and quantitatively.
MP3: Construct viable arguments and critique the reasoning of others.
MP4: Model with mathematics.
MP5: Use appropriate tools strategically.
MP6: Attend to precision.
MP7: Look for and make use of structure.
MP8: Look for and express regularity in repeated reasoning.
How often does the team that is winning at halftime go on to win the game? How unusual is it for a team to be winning at halftime to go on to lose the game? Below is the data that shows the halftime lead (𝑅𝑜𝑎𝑑 − 𝐻𝑜𝑚𝑒) and the final lead (Road – Home) for all 32 games that took place the first 2 weeks of the 2012 regular season.
HT Final
4 7
-3 -20
7 1
3 -4
18 21
HT Final
-21 -20
9 8
-7 -4
-13 -6
3 -12
HT Final HT Final
-7 -7 -6 -20
-11 -3 -3 -17
3 -22 -14 -28
0 2 -8 -8
11 -7 -13 -6
3 16
2 -3
-7 -31
4 8
10 -1
-8 -8
6 8
-20 -20
-13 -13
-21 -18
17 20
5 -3
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HCPSS Worthwhile Math Task
Data from: http://scores.espn.go.com/nfl/scoreboard?seasonYear=2012&seasonType=2&weekNumber=1 http://scores.espn.go.com/nfl/scoreboard?seasonYear=2012&seasonType=2&weekNumber=2
What does looking at this data for face value suggest about this question?
Create a scatter plot, using the data above, of the halftime lead versus the final lead. Describe the relationship you see.
What questions arise from this display of data? Would it be appropriate to formulate a linear model for this data to make predictions for future weeks? Why or why not?
1.
The intro to this lesson could be a discussion about various sports at first and then narrowing it down to football. This gives a chance for all students who like sports to chime in. For those that don’t follow (or have interest in) sports this would be a great opportunity to get those that do to explain some basics to help understand the scenario presented before starting.
2.
The teams that played are purposely left off during this first part so that hopefully there would be no bias towards a particular team. Not knowing this information will hopefully be brought up in the discussion the students have.
3.
For sake of time and the follow up questions, technology should be used to create this scatterplot by the students. There is no need to do this by hand, but a discussion of the labels necessary should occur. To help the discussion after students have completed the task, the teacher should have a scatterplot to be displayed while going over the answers.
1.
Find the linear model for this data then create a residual plot to determine if it is appropriate.
2.
Interpret the slope and the y-intercept of the LSRL within the context of the given problem.
3.
On 10/15/2012, the San Diego Chargers were beating the Denver Broncos 24-0 at halftime. Using the given model, by how much would we predict the Chargers to win by?
4.
The Chargers went on to lose that game by 24-35. What is the value of this residual?
Would you consider this data point an outlier if it was added to the scatterplot? If so, describe the scatterplot and the influence on the model.
5.
What other questions might you be able to ask about this data and/or model that would be valuable?
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Unported License .

HCPSS Worthwhile Math Task
Looking at the data for face value we see that only 8 out of the 32 games does the direction of who is winning at half time change by the end of the game. One might conclude that there is a relationship between these two variables. If a team is ahead at half time their chances are good that they will win the game.
Scatterplot:
This scatterplot appears to be a positive, linear form with moderate strength. There does not appear to be any clear outliers but there are some gaps that might be noted.
Sample questions that might arise: Who are these teams? What were the scores?
Looking at the scatterplot it appears that a linear model might be appropriate but we would need to look at the residual plot to be sure.
Follow Up Questions:
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HCPSS Worthwhile Math Task
1.
Scatterplot with LSRL: y-hat = 0.863x – 4.049 or predicted final lead = 0.863(half time lead) – 4.049
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
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Residual plot:
HCPSS Worthwhile Math Task
There does not appear to be a pattern in the residual plot therefore a linear model is appropriate.
2.
Slope: On average for each point the road team has over the home at half time they will have 0.863 points over them at the end of the game.
Y-intercept: At the start of the game the home team will have 4.049 points over the road team already. Hmmm….
3.
Assuming that the Chargers were the road team y

0.863(24)
  it would be by approximately 16 or 17 points. If they were the home team y

0.863( 24)

4.049
 
24.761
then is would be approximately 24 or 25 points.
4.
Again assuming they were the road team then the actual value was -11 and the model would have predicted 17. Therefore the residual would be -11 – 17 = -28. If they were the home team then the actual value was 11 and the model would have predicted 25 so the residual would be 11 – 25 = -14.
Value added to scatterplot with value circled for the first assumption:
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HCPSS Worthwhile Math Task
Summary stats with new point versus without the point:
This does appear to be an outlier. We note that the correlation and the slope go down with this new point so it appears to be an influential point. Since the value of the point and the mean of all the points are so different it has high leverage.
5.
Answers will vary.
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0
Unported License .