Testing for correlation Problem set A

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.
2. (b) A cyclist believes that the steeper a hill, the lower her average speed.
3. (c) A farmer asks whether there is any link between a month's rainfall and the 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.

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.

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.

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.
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 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.

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.
2. (b) Use your calculator to find r and the p-value.
3. (c) State your conclusion in context.

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.
2. (b) Test, at the 1% significance level, whether there is any 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.

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.
2. (b) Test, at the 5% significance level, whether there is any correlation.
3. (c) Explain why the two conclusions differ.