- Ranpose hospital offers services to a large number of clinics that refer patients to a range of hospitals.
The manager at Ranpose hospital took a random sample of 16 clinics and recorded
- the distance, \(x \mathrm {~km}\), of the clinic from Ranpose hospital
- the percentage, \(y \%\), of the referrals from the clinic who attend Ranpose hospital.
The data are summarised as
$$\bar { x } = 8.1 \quad \bar { y } = 20.5 \quad \sum y ^ { 2 } = 8266 \quad \mathrm {~S} _ { x x } = 368.16 \quad \mathrm {~S} _ { x y } = - 630.9$$
- Find the product moment correlation coefficient for these data.
- Give an interpretation of your correlation coefficient.
The manager at Ranpose hospital believes that there may be a linear relationship between the distance of a clinic from the hospital and the percentage of the referrals who attend the hospital. She drew the following scatter diagram for these data.
\includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-20_1106_926_1133_511} - State, giving a reason, whether or not these data support the manager's belief.
(1)
\section*{[The summary data and the scatter diagram are repeated below.]}
The data are summarised as
$$\bar { x } = 8.1 \quad \bar { y } = 20.5 \quad \sum y ^ { 2 } = 8266 \quad \mathrm {~S} _ { x x } = 368.16 \quad \mathrm {~S} _ { x y } = - 630.9$$
\includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-22_1118_936_612_504} - Find the equation of the regression line of \(y\) on \(x\), giving your answer in the form
$$y = a + b x$$
- Give an interpretation of the gradient of your regression line.
- Draw your regression line on the scatter diagram.
The manager believes that Ranpose hospital should be attracting an "above average" percentage of referrals from clinics that are less than 5 km from the hospital. She proposes to target one clinic with some extra publicity about the services Ranpose offers.
- On the scatter diagram circle the point representing the clinic she should target.
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