- Some students are investigating the strength of wire by suspending a weight at the end of the wire. They measure the diameter of the wire, \(d \mathrm {~mm}\), and the weight, \(w\) grams, when the wire fails. Their results are given in the following table.
| \cline { 2 - 13 }
\multicolumn{1}{l|}{} | These 14 points are plotted on page 13 | Not yet plotted | |
| \(d\) | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1.1 | 1.3 | 1.6 | 2 | 2.4 | 2.8 | 3.3 | 3.5 | 3.9 | \(\mathbf { 4 . 5 }\) | \(\mathbf { 4 . 6 }\) | \(\mathbf { 4 . 8 }\) | \(\mathbf { 5 . 4 }\) |
| \(w\) | 1.2 | 1.7 | 2.3 | 3.0 | 3.8 | 5.6 | 7.7 | 11.6 | 18 | 25.9 | 34.9 | 47.4 | 52.7 | 63.9 | \(\mathbf { 8 1 }\) | \(\mathbf { 8 3 . 6 }\) | \(\mathbf { 8 9 . 9 }\) | \(\mathbf { 1 0 9 . 4 }\) |
The first 14 points are plotted on the axes on page 13.
- On the axes on page 13, complete the scatter diagram for these data.
- Use your calculator to write down the equation of the regression line of \(w\) on \(d\).
- With reference to the scatter diagram, comment on the appropriateness of using this linear regression model to make predictions for \(w\) for different values of \(d\) between 0.5 and 5.4
The product moment correlation coefficient for these data is \(r = 0.987\) (to 3 significant figures).
- Calculate the residual sum of squares (RSS) for this model.
Robert, one of the students, suggests that the model could be improved and intends to find the equation of the line of regression of \(w\) on \(u\), where \(u = d ^ { 2 }\)
He finds the following statistics
$$\mathrm { S } _ { w u } = 5721.625 \quad \mathrm {~S} _ { u u } = 1482.619 \quad \sum u = 157.57$$ - By considering the physical nature of the problem, give a reason to support Robert's suggestion.
- Find the equation of the regression line of \(w\) on \(u\).
- Find the residual sum of squares (RSS) for Robert's model.
- State, giving a reason based on these calculations, which of these models better describes these data.
- Hence estimate the weight at which a piece of wire with diameter 3 mm will fail.
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