Spearman's rank correlation Problem set B One question at a time

A student records the minutes of practice each day and the score in a weekly quiz, over eight weeks. Calculate Spearman's rank correlation coefficient.

• Ranks of practice (min): 4, 6, 2, 8, 5, 1, 7, 3
• Ranks of score: 3, 6, 2, 8, 4, 5, 7, 1
• From the calculator: r_s = 0.738

Eight students record their minutes of daily revision and their mark in a short quiz. Calculate Spearman's rank correlation coefficient.

• Ranks of revision (min) (ties averaged): 2.5, 6.5, 4, 6.5, 1, 5, 2.5, 8
• Ranks of mark (ties averaged): 2.5, 7, 4, 5, 1, 7, 2.5, 7
• From the calculator: r_s = 0.907

Two film critics place eight films, A to H, in order, from 1st place to 8th place.

1. (a) Write down the rank each judge gave to A, B, C, D, E, F, G and H, in that order.

   • Find each letter in each list. Ranks of A to H for Critic 1: 3, 5, 7, 1, 8, 4, 2, 6
   • Ranks of A to H for Critic 2: 4, 5, 6, 2, 8, 3, 1, 7

2. (b) Calculate Spearman's rank correlation coefficient.

   • From the calculator: r_s = 0.929

A kiosk records the weekly hours of sunshine and the number of ice creams sold (tens) over seven weeks. Use Spearman's rank correlation to test, at the 5% significance level, whether there is a positive correlation. The critical value is 0.714.

1. (a) State the hypotheses.

   • H0: there is no correlation between hours of sunshine and ice cream sales
   • H1: there is a positive correlation between hours of sunshine and ice cream sales

2. (b) Calculate r_s.

   • Ranks of sunshine (h): 3, 5, 2, 7, 4, 6, 1
   • Ranks of sales: 3, 5, 2, 7, 6, 4, 1
   • From the calculator: r_s = 0.857

3. (c) State the conclusion in context.

   • r_s = 0.857 > 0.714, so reject H0.
   • There is sufficient evidence, at the 5% significance level, of a positive correlation between hours of sunshine and ice cream sales.

Eight students record the number of days they were absent (x) and their exam mark (y). Use Spearman's rank correlation to test, at the 10% significance level, whether there is a negative correlation. The critical value is 0.524.

1. (a) State the hypotheses.

   • H0: there is no correlation between days absent and exam mark
   • H1: there is a negative correlation between days absent and exam mark

2. (b) Calculate r_s.

   • Ranks of days absent: 2, 5, 3, 8, 4, 7, 1, 6
   • Ranks of mark: 7, 5, 6, 2, 4, 3, 8, 1
   • From the calculator: r_s = −0.905

3. (c) State the conclusion in context.

   • r_s = −0.905 < −0.524, so reject H0.
   • There is sufficient evidence, at the 10% significance level, of a negative correlation between days absent and exam mark.

A garage records the speed of nine test drives (km/h) and the fuel economy (miles per gallon). The critical value for a two-tailed test at the 5% significance level is 0.700.

1. (a) Calculate r_s.

   • Ranks of speed: 3, 6, 1, 9, 5, 7, 2, 4, 8
   • Ranks of economy: 8, 4, 9, 2, 6, 3, 7, 5, 1
   • From the calculator: r_s = −0.950

2. (b) Test, at the 5% significance level, whether there is any correlation. State the hypotheses and the conclusion in context.

   • H0: there is no correlation between speed and fuel economy. H1: there is a correlation between speed and fuel economy.
   • |r_s| = 0.950 > 0.700, so reject H0.
   • There is sufficient evidence, at the 5% significance level, of a correlation between speed and fuel economy.

Two coaches score six gymnasts out of 10.

1. (a) Calculate r_s.

   • Ranks of Coach 1: 3, 5, 2, 6, 4, 1
   • Ranks of Coach 2: 2, 6, 3, 4, 5, 1
   • From the calculator: r_s = 0.771

2. (b) The critical value for a one-tailed test at the 5% significance level is 0.829. Determine whether there is significant evidence that the two coaches tend to agree.

   • H0: there is no correlation between the two sets of scores. H1: there is a positive correlation.
   • r_s = 0.771 < 0.829, so do not reject H0.
   • There is insufficient evidence, at the 5% significance level, that the two coaches tend to agree.

Eight adults record their weekly exercise (hours) and their systolic blood pressure (mmHg).

1. (a) Calculate r_s and describe the correlation in context.

   • Ranks of exercise (h): 1, 6, 3, 8, 2, 7, 4, 5
   • Ranks of pressure: 8, 4, 5, 1, 7, 2, 6, 3
   • From the calculator: r_s = −0.952
   • strong negative correlation: people who exercise more tend to have lower blood pressure.

2. (b) A newspaper says that people who exercise more always have lower blood pressure. Does the data support this claim?

   • r_s = −0.952 is close to −1 but not equal to −1.
   • No. r_s is not exactly −1, so the ranks are not perfectly reversed: at least one person exercises more than another but has higher blood pressure. The data show a tendency, not 'always'.

The scatter diagram shows the mass y (kg) of a crop after x doses of fertiliser, for ten plots. For these data Pearson's r = 0.697 and Spearman's r_s = 0.964. The point (10, 160) was misrecorded. The correct value is (10, 45).

1. (a) Without calculating, say how the new value of Pearson's r compares with the old value.

   • The point was far from the pattern. Once it is moved onto the pattern, the points lie closer to a straight line, so r increases.

2. (b) Without calculating, say how the new value of r_s compares with the old value.

   • The point was the largest y before and is still the largest y after, so no rank changes and r_s does not change.

3. (c) The masses are converted from kilograms to grams. How does this affect r_s?

   • Multiplying every mass by 1000 keeps the order of the masses.
   • The ranks are unchanged, so r_s is unchanged.

The side length x (cm) and the volume y (cm³) of eight cubes are shown.

1. (a) Calculate r_s.

   • Ranks of side: 1, 2, 3, 4, 5, 6, 7, 8
   • Ranks of volume: 1, 2, 3, 4, 5, 6, 7, 8
   • From the calculator: r_s = 1

2. (b) Pearson's r for these data is 0.932. Explain why r_s = 1 although r is less than 1.

   • Every increase in x comes with an increase in y, so the ranks are identical and r_s = 1.
   • The points follow a curve, not a straight line, so Pearson's r is less than 1.