The Soccer
Scout Report

It turns out mathematicians already invented the number you described in Task 1b. It is called Pearson's product-moment correlation coefficient, denoted \(r\), and it is calculated as:

where the three sums are defined as:

\(S_{xx}\) measures how spread out the \(x\) values are around their mean. \(S_{yy}\) does the same for \(y\). \(S_{xy}\) measures how the two variables move together — it is large and positive when both variables tend to be above (or both below) their means at the same time.

Split the 12 players across your group members. Each person computes their rows, then reconvene to sum the columns and calculate \(r\). Check: does your answer have the properties your group listed in Task 1b?

Now let your calculator do it. Follow the steps for your model:

Does the GDC value of \(r\) match your hand calculation? Write one sentence interpreting \(r\) in context — what does it tell the scout about the relationship between training and sprint speed?

Calculate \(\bar{x}\) and \(\bar{y}\). Plot the mean point \((\bar{x}, \bar{y})\) on your scatter diagram. Draw a line of best fit by eye through this point. Then use your GDC to find the regression line of \(y\) on \(x\). Compare the GDC line to your sketch — how close were you?

Use the regression line of \(y\) on \(x\) to predict the sprint speed of a player who trains 10 hours per week. Show your substitution clearly.

Using the regression line from Phase 2, make three predictions and classify each one. For each: state whether it is an interpolation or extrapolation, and decide whether you trust the prediction. Justify your answer on the board.

• Predict sprint speed for a player training 9 hrs/wk
• Predict sprint speed for a player training 12 hrs/wk
• Predict sprint speed for a player training 25 hrs/wk

The scout wants to recruit a player who runs at exactly 7.6 m/s. How many training hours per week should that player be doing?

Step 1: Try answering using the regression line you already have. Rearrange it to make \(x\) the subject and substitute \(y = 7.6\). Write your method and answer on the board.

Step 2: Now use your GDC to find a second line — the regression line of \(x\) on \(y\). On the Casio: same CALC → REG menu, look for the \(x\) on \(y\) option. On the TI-Nspire: same Linear Regression but swap the roles of the columns (x-list = B, y-list = A). Use this new line to predict the training hours for a player running 7.6 m/s.

Step 3: Compare your two answers. Are they the same? Discuss with your group why you might get different results.

There is a regression line of \(y\) on \(x\), and a separate regression line of \(x\) on \(y\). Both pass through the mean point \((\bar{x}, \bar{y})\). Use your GDC to find the \(x\) on \(y\) line, plot both lines on the same diagram, and answer:

• Where do the two lines intersect?
• Are they the same line? If not, what is different about them?
• Which line should you use to answer Task 3b — and why?
• When would the two lines be identical?

A club sponsor has looked at this data and made a claim: "On matches when we sell more churros, the team scores more goals. We should invest in selling more churros to help the team win."

Before you agree or disagree, run the full analysis: calculate \(r\), find the regression line of goals on churros, and predict the number of goals if 600 churros are sold. Write all results on the board — do not draw any conclusions yet.

Now look critically at your results and discuss these questions on the board:

• Is your prediction for 600 churros an interpolation or an extrapolation? Does that affect how much you trust it?
• Think about when matches tend to have large crowds. What kinds of matches attract more fans?
• Would those same matches also tend to have more goals? Why?
• So what is actually driving both the churros sales and the goals — is it the churros themselves, or something else?
• If the club sells 600 churros at a quiet Tuesday league match, do you expect the prediction to hold?

Write a final recommendation on the board for the scout. Your verdict must include: the value of \(r\) and what it shows, whether you trust the prediction and why, and the real reason the two variables appear to be linked. Should the sponsor invest in churros?