- A scientist measured the salinity of water, \(x \mathrm {~g} / \mathrm { kg }\), and recorded the temperature at which the water froze, \(y ^ { \circ } \mathrm { C }\), for 12 different water samples. The summary statistics are listed below.
$$\begin{gathered}
\sum x = 504 \quad \sum y = - 27 \quad \sum x ^ { 2 } = 22842 \quad \sum y ^ { 2 } = 62.98
\sum x y = - 1190.7 \quad \mathrm {~S} _ { x x } = 1674 \quad \mathrm {~S} _ { y y } = 2.23
\end{gathered}$$
- Find the mean and variance of the recorded temperatures.
(3)
Priya believes that the higher the salinity of water, the higher the temperature at which the water freezes. - Calculate the product moment correlation coefficient between \(x\) and \(y\)
- State, with a reason, whether or not this value supports Priya's belief.
- Find the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\) Give the value of \(a\) and the value of \(b\) to 3 significant figures.
- Estimate the temperature at which water freezes when the salinity is \(32 \mathrm {~g} / \mathrm { kg }\)
The coding \(w = 1.8 y + 32\) is used to convert the recorded temperatures from \({ } ^ { \circ } \mathrm { C }\) to \({ } ^ { \circ } \mathrm { F }\)
- Find an equation of the least squares regression line of \(w\) on \(x\) in the form \(w = c + d x\)
- Find
- the variance of the recorded temperatures when converted to \({ } ^ { \circ } \mathrm { F }\)
- the product moment correlation coefficient between \(w\) and \(x\)
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