NCEA Level 1AS 91944RelationshipsNCEA Level 1 Maths

Scatter plots: 7 questions solved step by step

What this lesson covers

  • A scatter plot shows whether two numerical variables change together, and how closely.
  • An association between two variables does not show that one causes the other.
Key idea

7 questions, one data set

  • An Ōrewa dairy, 10 summer days.
  • Across: maximum temperature (°C).
  • Up: ice creams sold that day.
  • Answer in context, with units.
°CSold°CSold
18162425
19152612
20192734
22202833
23253037
Your turn

Question 1 · Achieved

  • Describe the trend in context.
  • Quote values from the data.
Question 1 · Achieved
Worked example

Q1: the trend

  • Read the whole cloud.
  • Say "tend": not every day.
  1. 18 °C: 16 ice creams sold
  2. 30 °C: 37 ice creams sold
  3. cloud rises left to rightbottom left to top right
  4. direction: positive
  5. Hotter days: sales tend to rise
Your turn

Question 2 · Achieved

  • Describe the strength and scatter.
  • Name any unusual point.
Question 2 · Achieved
Worked example

Q2: strength and scatter

  • Strong: close to a line.
  • Unusual: keep it, explain.
  1. 9 of 10 dots near a straight line
  2. strength: strong; form: linear
  3. scatter: even from 18 to 30 °C
  4. unusual: 26 °C, only 12 sold
  5. 24 °C: 25 sold; 27 °C: 34 sold
  6. 12 is less than half of 25
  7. keep it: rain? closed early?
  8. Strong, linear, 1 unusual day
Your turn

Question 3 · Merit

  • A trend line drawn by eye passes
  • through (18, 14) and (30, 38).
  • Find its equation. What does
  • the gradient mean in context?
Question 3 · Merit
Worked example

Q3: the trend line

  • Gradient = rise ÷ run.
  • Then substitute for c.
  1. m = (38 − 14) ÷ (30 − 18)rise ÷ run
  2. = 24 ÷ 12 = 2
  3. y = 2x + c
  4. 14 = 2 × 18 + cuse (18, 14)
  5. 14 = 36 + c, so c = −22
  6. y = 2x − 22
  7. 2 × 30 − 22 = 38the other point
  8. about 2 more sold per extra 1 °C
  9. c = −22: no meaning at 0 °C
Your turn

Question 4 · Merit

  • Use y = 2x − 22 to predict sales
  • on a 25 °C day.
  • Interpolation or extrapolation?
  • How reliable is it?
Worked example

Q4: predict for 25 °C

  • Substitute x = 25.
  • Then check the range.
  1. y = 2 × 25 − 22
  2. = 50 − 22 = 28
  3. data: 18 °C to 30 °C
  4. 25 °C is inside: interpolation
  5. strong, dots close to the line
  6. but only 10 days, one dairy
  7. About 28 ice creams sold
Your turn

Question 5 · Merit, in context

  • A winter day has a maximum of 6 °C.
  • Use y = 2x − 22 to predict sales.
  • Does the prediction make sense?
Worked example

Q5: predict for 6 °C

  • Calculate, then question it.
  • No winter days in the data.
  1. y = 2 × 6 − 22
  2. = 12 − 22 = −10
  3. −10 ice creamssales cannot be negative
  4. 6 °C is outside 18 to 30 °C
  5. extrapolation: no winter data
  6. The line cannot predict winter
Your turn

Question 6 · Excellence

  • The owner says the 26 °C day
  • proves the trend line is useless.
  • Use the line to evaluate her claim.
Question 6 · Excellence
Worked example

Q6: is the line useless?

  • Predicted versus actual.
  • Compare the other days.
  1. 2 × 26 − 22 = 30predicted
  2. 12 − 30 = −18actual − predicted
  3. other 9 days: within 2 of line
  4. 24 °C: line 26, sold 25
  5. 27 °C: line 32, sold 34
  6. 1 day far off, 9 days close
  7. "The line is useless"fits 9 days of 10
  8. The line fits normal days well
Your turn

Question 7 · Excellence

  • A medical centre near the beach saw
  • more sunburn on high-sales days.
  • "Ice creams cause sunburn."
  • Evaluate the claim.
Worked example

Q7: association or cause?

  • Look for a third variable.
  • Associated is not caused.
  1. sales and sunburn rise together
  2. that is an association
  3. third variable: hot, sunny days
  4. sun means more ice creams sold
  5. sun means more sunburn cases
  6. "Ice creams cause sunburn"not shown by this data
  7. observational, not an experiment
  8. Associated with, not caused by
Summary

What answered all seven

  • Describe, fit, predict, judge, and never claim a cause.
StepHow
Directionpositive, negative or none
Strengthhow close dots are to a line
Unusualname it, keep it, give a reason
Linegradient first, then c
Predictinside the data: interpolate
Outsideextrapolation: do not trust
Causesay "associated with"

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