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

A gardener gives each of 7 plants a mass of fertiliser x (g) and measures the yield y (kg) of tomatoes. x: 2, 4, 5, 7, 8, 10, 11 y: 3.1, 3.8, 4.6, 5, 5.9, 6.2, 7.4

1. (a) Find the mean point.

   • From the calculator: \(\bar{x}=6.71\), \(\bar{y}=5.14\)
   • Mean point: (6.71, 5.14)

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 (6.71, 5.14) in a different colour
   • Draw a line through the mean point with about the same number of points on each side

For 8 teenagers, the daily screen time x (hours) and the hours of sleep y are recorded. x: 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5 y: 8.6, 8.1, 8.3, 7.4, 7.6, 6.9, 7.1, 6.4

1. (a) Find Pearson's r.

   • From the calculator: \(r=-0.948\)

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

   • r is negative: as screen time increases, sleep decreases
   • \(|r|=0.948\) is close to 1: the points lie close to a line
   • Strong negative linear correlation

For 12 office workers, the minutes x spent commuting each day and a wellbeing score y (out of 100) are recorded. x: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 y: 62, 58, 66, 49, 55, 47, 40, 52, 36, 41, 30, 38

1. (a) Find r.

   • From the calculator: \(r=-0.873\)

2. (b) For n = 12 the critical value of r is 0.576. Is there significant evidence of correlation?

   • Compare the size of r: \(|r|=0.873\)
   • \(0.873>0.576\): |r| is larger than the critical value
   • Yes: significant evidence of negative correlation (strong negative correlation)

A tank drains. The scatter diagram shows the water depth y (m) at 12 readings, taken x hours after noon. Which of these values is closest to r: 0.95, 0.5, 0.05 or −0.95?

• The points fall from left to right, so r is negative
• The points lie close to a straight line, so |r| is close to 1
• From the calculator: \(r=-0.949\), closest to −0.95

At a seaside kiosk, over 9 days, the number of ice lollies sold x (tens) and the number y of sunburn cases treated at the nearby first-aid post are recorded. x: 2, 3, 4, 5, 6, 7, 8, 9, 10 y: 14, 17, 17, 24, 27, 28, 35, 36, 41

1. (a) Find r.

   • From the calculator: \(r=0.988\)
   • Strong positive linear correlation

2. (b) A reporter writes: "Eating ice lollies causes sunburn." 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
   • Hot, sunny weather makes people buy lollies and also makes them burn
   • So the lurking variable is the weather

Ten employees work for two organisations. For each, the years of experience x and the salary y (€ thousand) are recorded. The first five are at a city bank (Firm 1) and the last five at a village charity (Firm 2). x: 3, 4, 5, 6, 7, 9, 10, 11, 12, 13 y: 71, 74, 73, 78, 80, 40, 43, 41, 47, 48

1. (a) Find r for all ten.

   • From the calculator: \(r=-0.807\)
   • Negative: the ten points taken together seem to fall

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

   • Firm 1: \(r=0.940\)
   • Firm 2: \(r=0.887\)
   • Both are strong and positive

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

   • Within each firm the correlation is positive: more experience goes with a higher salary
   • Firm 2 staff have more experience but much lower salaries, which pulls the combined r below 0
   • The firm is a lurking variable: the ten points should not be treated as one group

A candle is lit and its height y (cm) is measured x hours later. x: 4, 5, 7, 8, 9, 11, 13, 15, 18 y: 41, 31, 30, 28, 23, 19, 21, 16, 7 A line of best fit is drawn by eye through the mean point and the point (16, 12).

1. (a) Find the mean point.

   • From the calculator: \(\bar{x}=10\), \(\bar{y}=24\)

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

   • \(m=\frac{12-24}{16-10}=-2\)
   • \(y-24=-2(x-10)\)
   • \(y=-2x+44\)

3. (c) Use the line to estimate the height of the candle after 6 hours.

   • \(y=-2(6)+44=32\)
   • The candle is about 32 cm tall after 6 hours.

Two reservoirs are checked on days 1 to 9 after heavy rain. The depth y (m) of reservoir P and of reservoir Q on day x is: x: 1, 2, 3, 4, 5, 6, 7, 8, 9 P: 11.2, 9.7, 9.2, 7.5, 6.7, 4.3, 5.5, 4.1, 4.1 Q: 9.6, 9.4, 9, 8.5, 7.7, 6.8, 5.4, 3.9, 2

1. (a) Find r and the mean of y for each set.

   • From the calculator, set P: \(r=-0.959\), \(\bar{y}=6.92\)
   • Set Q: \(r=-0.959\), \(\bar{y}=6.92\)
   • 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 ball is thrown over a wall. At horizontal distances x (m) from the highest point, its height y (m) is: x: −3, −2, −1, 0, 1, 2, 3 y: 0, 5, 8, 9, 8, 5, 0

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 = 0, so the line of best fit is flat and r = 0
   • But y = 9 − x² for every point: a perfect curved relationship
   • r = 0 means no linear correlation, not no relationship