A health researcher records the daily step count, x x\,x thousand steps, and the resting heart rate, y y\,y beats per minute, of each of 200 adults. For the full sample the product moment correlation coefficient between x x\,x and y y\,y is -0.62.
The researcher repeats the calculation using only the 40 adults whose daily step count lies between 9 and 11 thousand steps, and obtains a product moment correlation coefficient of -0.08. Explain why restricting the sample in this way reduces the magnitude of the correlation coefficient, even though the relationship in the full sample has not changed.
The researcher concludes that walking more causes a person's resting heart rate to fall. Give one reason why her data do not establish this, and name a variable that could produce the association without any such causal link.
A colleague argues that because -0.62 is not close to -1, a linear model is of no use for these data. Explain why this argument is not sound.
State one further piece of information about the 200 adults that the researcher would need before using a regression line of y y\,y on x x\,x to predict the resting heart rate of an adult who walks 25 thousand steps a day.
244 exam-style questions on AQA A Level Maths 2.2 L: Data presentation and interpretation, covering 2.2.1 Single-variable data diagrams, 2.2.2 Scatter diagrams and correlation, 2.2.3 Central tendency, variation and standard deviation, 2.2.4 Outliers and cleaning data, and 2.2 L: Data presentation and interpretation. Each one has a worked solution and a mark scheme showing where the marks go.