Testing for correlation Problem set A Worked solutions

Write down H0 and H1 for each test. Define ρ in each case.

1. (a) A teacher believes that pupils who read more pages per week score higher in a vocabulary test.

   • H0 is always ρ = 0. More of one goes with more of the other: positive, H1: ρ > 0.
   • H0: ρ = 0, H1: ρ > 0. ρ is the population correlation coefficient between pages read per week and vocabulary mark.

2. (b) A cyclist believes that the steeper a hill, the lower her average speed.

   • H0 is always ρ = 0. One rises as the other falls: negative, H1: ρ < 0.
   • H0: ρ = 0, H1: ρ < 0. ρ is the population correlation coefficient between steepness of a hill and average cycling speed.

3. (c) A farmer asks whether there is any link between a month's rainfall and the milk yield per cow.

   • H0 is always ρ = 0. Any link, in either direction: H1: ρ ≠ 0.
   • H0: ρ = 0, H1: ρ ≠ 0. ρ is the population correlation coefficient between monthly rainfall and milk yield per cow.

A shop records the hours of sunshine and the number of ice creams sold on eight days. Sunshine (hours): 3.5, 4.0, 5.5, 6.0, 7.5, 8.0, 9.5, 10.0; Ice creams sold (hundreds): 8.7, 6.3, 9.4, 6.6, 9.7, 10.5, 8.7, 10.8. Use your calculator to find r and the p-value for a test with H1: ρ > 0.

• From the calculator: linear regression t test with H1: ρ > 0.
• r = 0.615
• p = 0.0524

A geographer records the height above sea level and the average annual temperature of seven villages. Height (hundreds of m): 2, 5, 9, 14, 18, 23, 28; Temperature (°C): 8.0, 13.2, 9.4, 8.1, 9.7, 6.9, 4.5. Test, at the 10% significance level, whether there is negative correlation between height above sea level and average temperature.

• H0: ρ = 0, H1: ρ < 0. ρ is the population correlation coefficient between height above sea level and average temperature.
• From the calculator: linear regression t test with H1: ρ < 0: r = −0.694, p = 0.0417
• p = 0.0417 < 0.1, so reject H0. There is sufficient evidence at the 10% level of negative correlation between height above sea level and average temperature.

A librarian records the length of nine novels and the average rating given by readers. Length (pages): 180, 220, 260, 310, 350, 420, 480, 550, 640; Rating (out of 10): 9.0, 4.4, 7.4, 6.4, 6.5, 6.8, 5.0, 6.8, 6.1. Test, at the 5% significance level, whether there is any correlation between the length of a novel and the reader rating.

• H0: ρ = 0, H1: ρ ≠ 0. ρ is the population correlation coefficient between the length of a novel and the reader rating.
• From the calculator: linear regression t test with H1: ρ ≠ 0: r = −0.276, p = 0.472
• p = 0.472 > 0.05, so do not reject H0. There is insufficient evidence at the 5% level of correlation between the length of a novel and the reader rating.

Answer these questions about the p-value of a test for correlation.

1. (a) A test for correlation gives a p-value. State the assumption about the variables that must hold for this p-value to be valid.

   • The p-value is found from r and the number of pairs.
   • It assumes that both variables come from normal populations.

2. (b) A scatter diagram of heights against masses of 40 adults looks like an elliptical cloud. Explain why this supports using the p-value test.

   • A bivariate normal distribution gives an elliptical cloud of points.
   • The cloud has that shape here, so the assumption looks reasonable.

A school records the weekly hours of homework done by ten students and their percentage score in an end-of-term test. Homework (hours per week): 2, 3, 4, 5, 5.5, 6, 7, 8, 9, 10; Test score (%): 60, 36, 57, 54, 56, 59, 51, 65, 64, 92. Test, at the 2% significance level, whether more homework is linked to a higher score. State your hypotheses and your conclusion.

• H0: ρ = 0, H1: ρ > 0. ρ is the population correlation coefficient between homework time and test percentage.
• From the calculator: linear regression t test with H1: ρ > 0: r = 0.692, p = 0.0133
• p = 0.0133 < 0.02, so reject H0. There is sufficient evidence at the 2% level of positive correlation between homework time and test percentage.

A fitness app records the weekly training sessions and the resting heart rate of eight users. Sessions per week: 0, 1, 2, 3, 3, 4, 5, 6; Resting heart rate (bpm): 73, 70, 70, 71, 67, 63, 62, 66. A trainer claims that more sessions go with a lower resting heart rate. Test this claim at the 5% significance level.

1. (a) State suitable hypotheses, defining ρ.

   • A lower heart rate with more sessions is negative correlation.
   • H0: ρ = 0, H1: ρ < 0. ρ is the population correlation coefficient between sessions per week and resting heart rate.

2. (b) Use your calculator to find r and the p-value.

   • From the calculator: linear regression t test with H1: ρ < 0: r = −0.802, p = 0.00832

3. (c) State your conclusion in context.

   • 0.00832 < 0.05
   • p = 0.00832 < 0.05, so reject H0. There is sufficient evidence at the 5% level of negative correlation between sessions per week and resting heart rate.

A promoter records the ticket price and the audience size at nine concerts. Ticket price (€): 15, 20, 25, 30, 35, 40, 45, 50, 60; Audience (hundreds): 18.0, 16.7, 19.4, 21.1, 22.8, 20.3, 18.6, 18.0, 21.9.

1. (a) Find the correlation coefficient r.

   • From the calculator: linear regression t test gives r.
   • r = 0.407

2. (b) Test, at the 1% significance level, whether there is any correlation between ticket price and audience size.

   • H0: ρ = 0, H1: ρ ≠ 0. ρ is the population correlation coefficient between ticket price and audience size.
   • From the calculator: linear regression t test with H1: ρ ≠ 0: r = 0.407, p = 0.277
   • p = 0.277 > 0.01, so do not reject H0. There is insufficient evidence at the 1% level of correlation between ticket price and audience size.

A museum manager believes that the more the museum charges for entry, the fewer visitors it gets. She records the entry price and the number of visitors per day on eight days. Entry price (€): 4, 5, 6, 7, 8, 9, 10, 12; Visitors per day (hundreds): 13.5, 5.8, 12.4, 11.2, 13.3, 12.0, 14.4, 11.9. Test the belief at the 5% significance level, and comment on the sign of r.

• H0: ρ = 0, H1: ρ < 0. ρ is the population correlation coefficient between entry price and visitors.
• From the calculator: linear regression t test with H1: ρ < 0: r = 0.334, p = 0.790
• 0.790 > 0.05, so do not reject H0. There is insufficient evidence at the 5% level of negative correlation.
• r = 0.334 is positive: the sample shows the opposite trend, which is why p is so large.

A researcher records the daily exercise and the sleep of ten adults. Exercise (minutes): 10, 20, 30, 40, 45, 60, 75, 90, 100, 120; Sleep (hours): 6.4, 6.9, 6.6, 5.4, 7.3, 7.1, 6.5, 7.5, 7.5, 7.7.

1. (a) Test, at the 5% significance level, whether there is positive correlation.

   • From the calculator: linear regression t test with H1: ρ > 0: r = 0.626, p = 0.0265
   • 0.0265 < 0.05: reject H0. Sufficient evidence of positive correlation.

2. (b) Test, at the 5% significance level, whether there is any correlation.

   • From the calculator: linear regression t test with H1: ρ ≠ 0: r = 0.626, p = 0.0530
   • 0.0530 > 0.05: do not reject H0. Insufficient evidence of correlation.

3. (c) Explain why the two conclusions differ.

   • The two-tailed p-value is twice the one-tailed value: 0.0530 = 2 × 0.0265.
   • The first test commits to a direction before seeing the data. The second does not, so it asks for more.