3. Martin is investigating the relationship between a person's daily caffeine consumption, \(c\) milligrams, and the amount of sleep they get, \(h\) hours, per night. He collected this information from 20 people and the results are summarised below.
$$\begin{array} { c c }
\sum c = 3660 \quad \sum h = 126 \quad \sum c ^ { 2 } = 973228
\sum c h = 20023.4 \quad S _ { c c } = 303448 \quad S _ { c h } = - 3034.6
\end{array}$$
Martin calculates the product moment correlation coefficient for these data and obtains - 0.833
- Give a reason why this value supports a linear relationship between \(c\) and \(h\)
The amount of sleep per night is the response variable.
- Explain what you understand by the term 'response variable'.
Martin says that for each additional 100 mg of caffeine consumed, the expected number of hours of sleep decreases by 1
- Determine, by calculation, whether or not the data support this statement.
- Use the data to calculate an estimate for the expected number of hours of sleep per night when no caffeine is consumed.