Scatter diagrams and correlation Problem set A

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

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

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

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.
2. (b) A reporter writes: "Eating ice lollies causes sunburn." Explain why the data do not justify this, and name a lurking variable.

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.
2. (b) Find r for each firm on its own.
3. (c) Explain why the result in part (a) is misleading.

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.
2. (b) Find the equation of the line.
3. (c) Use the line to estimate the height of the candle 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.
2. (b) Plot both sets. Explain why r alone is misleading here.

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