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IB Mathematics: Applications and Interpretation HL · 13.3: Statistics · 13.3.6

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Scatter diagrams and correlation Problem set B

Use the calculator for the mean point and r. Give r to 3 significant figures, and plot the data whenever you can.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A used-car dealer records the age x (years) and the value y (€ thousand) of 7 cars. x: 1, 2, 4, 5, 7, 8, 10 y: 21.5, 19, 16.2, 14.8, 11.9, 10.1, 7.7

    1. (a) Find the mean point.
    2. (b) Draw a scatter diagram, mark the mean point and draw a line of best fit by eye.

  2. For 8 days, the outdoor temperature x (°C) at 7 a.m. and the energy y (kWh) used to heat a house that day are recorded. x: 4, 7, 9, 12, 15, 17, 20, 22 y: 31, 27, 26, 20, 18, 13, 12, 6

    1. (a) Find Pearson's r.
    2. (b) Describe the correlation between x and y.

  3. For 8 shoppers, the time x (minutes) spent in a shop and the amount y (€) spent are recorded. x: 2, 4, 5, 7, 8, 10, 11, 13 y: 14, 9, 17, 12, 18, 11, 19, 15

    1. (a) Find r.
    2. (b) For n = 8 the critical value of r is 0.707. Is there significant evidence of correlation?

  4. The scatter diagram shows the height y (m) of 12 trees against their age x (years). Which of these values is closest to r: 0.85, 0.3, −0.3 or −0.85?

    12345678910111213246810x (years)y (m)

In context

  1. For 9 fires, the number of fire engines sent x and the damage y (€ thousand) are recorded. x: 1, 2, 2, 3, 4, 5, 6, 8, 9 y: 8, 12, 20, 18, 35, 31, 48, 60, 66

    1. (a) Find r.
    2. (b) A reporter writes: "Sending more fire engines causes more damage." Explain why the data do not justify this, and name a lurking variable.

  2. Ten hotels are in two resorts. For each, the price x (€ per night) and the guest satisfaction score y (out of 50) are recorded. The first five are in a mountain village (Resort 1) and the last five at a beach (Resort 2). x: 20, 22, 24, 26, 28, 36, 38, 40, 42, 44 y: 34, 36, 35, 39, 41, 12, 15, 14, 18, 19

    15202530354045501020304050x (€)y (score)
    1. (a) Find r for all ten.
    2. (b) Find r for each resort on its own.
    3. (c) Explain why the result in part (a) is misleading.

  3. A rainwater tank fills during a storm. The volume y (litres) in the tank is measured x hours after the rain starts. x: 4, 6, 7, 11, 12, 14, 16, 18, 20 y: 12, 26, 29, 34, 40, 47, 46, 60, 66 A line of best fit is drawn by eye through the mean point and the point (20, 64).

    246810121416182022242610203040506070x (hours)y (litres)mean point(20, 64)
    1. (a) Find the mean point.
    2. (b) Find the equation of the line.
    3. (c) Use the line to estimate the volume in the tank after 9 hours.

Stretch

  1. Two bacterial cultures are photographed on days 1 to 9. The area y (cm²) covered by culture P and by culture Q on day x is: x: 1, 2, 3, 4, 5, 6, 7, 8, 9 P: 0.5, 1.5, 1.7, 0.8, 6, 7.3, 6.4, 8.8, 11.4 Q: 1.2, 1.5, 1.9, 2.6, 3.5, 4.9, 6.8, 9.4, 12.6

    1. (a) Find r and the mean of y for each set.
    2. (b) Plot both sets. Explain why r alone is misleading here.

  2. A theatre tries seven ticket prices x (€). The revenue y (€ hundreds) from each is: x: 2, 4, 6, 8, 10, 12, 14 y: 28, 48, 60, 64, 60, 48, 28

    1. (a) Find r.
    2. (b) Does r = 0 mean that there is no relationship between x and y? Explain.

jmaths.xyz · AI HL · 13.3.6 Scatter diagrams and correlation