Testing for correlation Problem set B

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

1. (a) A gardener believes that the more hours of sunshine a week, the more tomatoes she harvests.
2. (b) A café owner believes that the warmer the day, the fewer cups of hot chocolate he sells.
3. (c) An analyst asks whether there is any link between the population of a town and its number of libraries.

A restaurant records the number of staff on shift and the customers served per hour on eight evenings. Staff on shift: 3, 4, 5, 6, 7, 8, 9, 10; Customers served per hour: 41, 58, 34, 71, 62, 61, 67, 70. Use your calculator to find r and the p-value for a test with H1: ρ > 0.

A park manager records the altitude of seven mountain huts and the number of days per year that the path to each is open. Altitude (hundreds of m): 8, 10, 12, 15, 17, 20, 24; Days open per year: 304, 163, 246, 197, 210, 160, 160. Test, at the 10% significance level, whether there is negative correlation between the altitude of a hut and the days the path is open.

A cinema records the length of nine films and the average audience score. Length (minutes): 85, 92, 98, 105, 112, 120, 134, 150, 168; Audience score (%): 78, 42, 65, 57, 58, 60, 46, 60, 55. Test, at the 5% significance level, whether there is any correlation between the length of a film and the audience score.

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

1. (a) A student finds a p-value for the correlation between two measurements. State the assumption about the two variables that the p-value depends on.
2. (b) The marks of 30 students in a test are strongly skewed to the right. Explain what this means for a correlation test using these marks.

A music teacher records the weekly practice time of ten pupils and their mark in a grade exam. Practice (hours per week): 1, 2, 3, 3.5, 4, 5, 6, 7, 8, 9; Exam mark (out of 100): 48, 54, 68, 64, 52, 65, 65, 73, 66, 81. Test, at the 2% significance level, whether more practice is linked to a higher mark. State your hypotheses and your conclusion.

An agent records the distance from the city centre and the monthly rent of eight flats. Distance (km): 1.0, 2.5, 3.0, 4.5, 6.0, 7.5, 9.0, 11.0; Rent (€ per month): 1357, 1304, 1312, 1389, 1167, 1108, 1105, 1185. The agent claims that rent falls as the distance from the centre grows. 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 shop records the price of nine cycle helmets and the average rating given by customers. Price (€): 20, 25, 30, 35, 45, 55, 65, 80, 95; Rating (out of 5): 3.8, 3.1, 2.9, 2.3, 3.4, 4.0, 3.5, 3.7, 3.9.

1. (a) Find the correlation coefficient r.
2. (b) Test, at the 1% significance level, whether there is any correlation between price of a cycle helmet and customer rating.

A baker believes that the more she charges for a cake, the fewer cakes she sells. She records the price and the number sold per day for eight cakes. Price (€): 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.5; Cakes sold per day: 45, 11, 40, 34, 44, 38, 48, 37. Test the belief at the 5% significance level, and comment on the sign of r.

A librarian records the weekly time that ten students spend in the library and their exam marks. Library time (hours): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10; Exam mark (%): 54, 49, 52, 35, 75, 71, 60, 71, 69, 63.

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.