Scatter diagrams and correlation Problem set B One question at a time

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.

   • From the calculator: \(\bar{x}=5.29\), \(\bar{y}=14.5\)
   • Mean point: (5.29, 14.5)

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

   • Plot the 7 points and label both axes
   • Mark (5.29, 14.5) in a different colour
   • Draw a line through the mean point with about the same number of points on each side

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.

   • From the calculator: \(r=-0.989\)

2. (b) Describe the correlation between x and y.

   • r is negative: as the temperature increases, less energy is used
   • \(|r|=0.989\) is close to 1: the points lie close to a line
   • Strong negative linear correlation

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.

   • From the calculator: \(r=0.307\)

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

   • Compare the size of r: \(|r|=0.307\)
   • \(0.307<0.707\): |r| is smaller than the critical value
   • No: not significant evidence of correlation (weak positive correlation)

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?

• The points rise from left to right, so r is positive
• The points follow a line quite closely, so |r| is large but not 1
• From the calculator: \(r=0.846\), closest to 0.85

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.

   • From the calculator: \(r=0.981\)
   • Strong positive linear correlation

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.

   • The correlation shows that the two variables go up together. It does not show that one causes the other
   • A bigger fire needs more engines and also does more damage
   • So the lurking variable is the size of the fire

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

1. (a) Find r for all ten.

   • From the calculator: \(r=-0.844\)
   • Negative: the ten points taken together seem to fall

2. (b) Find r for each resort on its own.

   • Resort 1: \(r=0.922\)
   • Resort 2: \(r=0.933\)
   • Both are strong and positive

3. (c) Explain why the result in part (a) is misleading.

   • Within each resort the correlation is positive: a higher price goes with higher satisfaction
   • Resort 2 has higher prices but much lower satisfaction, which pulls the combined r below 0
   • The resort is a lurking variable: the ten points should not be treated as one group

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).

1. (a) Find the mean point.

   • From the calculator: \(\bar{x}=12\), \(\bar{y}=40\)

2. (b) Find the equation of the line.

   • \(m=\frac{64-40}{20-12}=3\)
   • \(y-40=3(x-12)\)
   • \(y=3x+4\)

3. (c) Use the line to estimate the volume in the tank after 9 hours.

   • \(y=3(9)+4=31\)
   • The tank holds about 31 litres after 9 hours.

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.

   • From the calculator, set P: \(r=0.941\), \(\bar{y}=4.93\)
   • Set Q: \(r=0.941\), \(\bar{y}=4.93\)
   • The summary values are the same

2. (b) Plot both sets. Explain why r alone is misleading here.

   • Plot P: the points scatter about a straight line
   • Plot Q: the points follow a curve, not a line
   • The same r and mean y do not mean the same pattern: r measures only linear correlation
   • Always plot the data before relying on r

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.

   • From the calculator: \(r=0\)

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

   • The points are symmetrical about x = 8, so the line of best fit is flat and r = 0
   • But y = x(16 − x) for every point: a perfect curved relationship
   • r = 0 means no linear correlation, not no relationship