For each: describe the relationship (direction, strength, pattern). Then rank all six from strongest to weakest relationship. Justify each rank in one sentence.
If you had to create a single number to measure relationship strength, what properties would it need? Write your answer on the board.
One plot would give a misleading result for any linear measure. Which one, and why?
| x (hrs/wk) | 4 | 5 | 6 | 6 | 7 | 8 | 8 | 9 | 10 | 11 | 12 | 13 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| y (m/s) | 6.1 | 6.4 | 6.3 | 6.8 | 7.0 | 7.2 | 6.9 | 7.5 | 7.4 | 7.8 | 7.9 | 8.1 |
\(S_{xx}=\sum(x_i-\bar x)^2\) · \(S_{yy}=\sum(y_i-\bar y)^2\) · \(S_{xy}=\sum(x_i-\bar x)(y_i-\bar y)\)
Split the 12 rows across your group, sum the columns, substitute. Does your answer match the properties from Task 1b?
Casio fx-CG50: STAT → CALC (F2) → REG (F3) → ax+b (F1). Read \(r\).
TI-Nspire: Lists & Spreadsheet → Menu → Statistics → Linear Regression (mx+b). Read \(r\).
Write one sentence interpreting \(r\) in context for the scout.
Find \(\bar x\) and \(\bar y\). Plot \((\bar x, \bar y)\) and sketch a line of best fit through it. Then find the regression line of \(y\) on \(x\) with your GDC. Why must the line pass through the mean point?
Predict the sprint speed of a player training 10 hrs/wk. Show the substitution.
For each: state interpolation or extrapolation, and whether you trust it. Justify on the board.
The scout wants a player who runs 7.6 m/s. Step 1: rearrange the y-on-x line and substitute. Step 2: find the x on y line via GDC (swap list roles). Step 3: compare both answers — are they the same?
Use your GDC to find the regression line of \(x\) on \(y\). Plot both lines on the same diagram and answer on the board:
| Match | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Churros (x) | 210 | 185 | 340 | 290 | 155 | 410 | 375 | 230 | 460 | 310 |
| Goals (y) | 1 | 0 | 3 | 2 | 1 | 4 | 3 | 1 | 4 | 2 |
| Match | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|
| Churros (x) | 270 | 195 | 385 | 440 | 160 | 325 | 280 | 500 | 215 | 355 |
| Goals (y) | 2 | 1 | 3 | 4 | 0 | 3 | 2 | 5 | 1 | 3 |
"On matches when we sell more churros, the team scores more goals. We should sell more churros to help the team win."
Find \(r\), the regression line of goals on churros, and predict goals if 600 churros are sold. Write results on the board — no conclusions yet.
Discuss on the board: Is 600 churros interpolation or extrapolation? When do matches attract big crowds — and would those same matches tend to have more goals? What is actually driving both variables? Would the prediction hold for a quiet Tuesday match?
Write a final recommendation. Include: the value of \(r\), whether the prediction is reliable, and the real reason the two variables appear to be linked. Should the sponsor invest in churros?