8 An experiment was conducted to see whether there was any relationship between the maximum tidal current, \(y \mathrm {~cm} \mathrm {~s} ^ { - 1 }\), and the tidal range, \(x\) metres, at a particular marine location. [The tidal range is the difference between the height of high tide and the height of low tide.] Readings were taken over a period of 12 days, and the results are shown in the following table.
| \(x\) | 2.0 | 2.4 | 3.0 | 3.1 | 3.4 | 3.7 | 3.8 | 3.9 | 4.0 | 4.5 | 4.6 | 4.9 |
| \(y\) | 15.2 | 22.0 | 25.2 | 33.0 | 33.1 | 34.2 | 51.0 | 42.3 | 45.0 | 50.7 | 61.0 | 59.2 |
$$\left[ \Sigma x = 43.3 , \Sigma y = 471.9 , \Sigma x ^ { 2 } = 164.69 , \Sigma y ^ { 2 } = 20915.75 , \Sigma x y = 1837.78 . \right]$$
The scatter diagram below illustrates the data.
\includegraphics[max width=\textwidth, alt={}, center]{2fb25fc5-0445-44fa-a23e-647d14b1a376-4_462_793_1464_644}
- Calculate the product moment correlation coefficient for the data, and comment briefly on your answer with reference to the appearance of the scatter diagram.
- Calculate the equation of the regression line of maximum tidal current on tidal range.
- Estimate the maximum tidal current on a day when the tidal range is 4.2 m , and comment briefly on how reliable you consider your estimate is likely to be.
- It is suggested that the equation found in part (ii) could be used to predict the maximum tidal current on a day when the tidal range is 15 m . Comment briefly on the validity of this suggestion.