Linearize non-linear relationships

2 Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C \left( a ^ { x } \right)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\). By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures.
\includegraphics[max width=\textwidth, alt={}, center]{21878d10-7f16-4dbb-86ef-65a7ba5eeafb-03_759_944_749_596}

1 A manufacturer of playground safety tiles is testing a new type of tile. Tiles of various thicknesses are tested to estimate the maximum height at which people would be unlikely to sustain injury if they fell onto a tile. The results of the test are as follows.

1. Draw a scatter diagram to illustrate these data.
2. State which of the two variables is the independent variable, giving a reason for your answer.
3. Calculate the equation of the regression line of maximum height on thickness.
4. Use the equation of the regression line to calculate estimates of the maximum height for thicknesses of
   (A) 70 mm ,
   (B) 120 mm . Comment on the reliability of each of these estimates.
5. Calculate the value of the residual for the data point at which \(t = 40\).
6. In a further experiment, the manufacturer tests a tile with a thickness of 200 mm and finds that the corresponding maximum height is 2.96 m . What can be said about the relationship between tile thickness and maximum height?

6. A missile was fired vertically upwards and its height above ground level, \(h\) metres, was found at various times \(t\) seconds after it was released. The results are given in the following table: It is thought that this data can be fitted to the formula \(h = p t - q t ^ { 2 }\).

1. Show that this equation can be written as \(\frac { h } { t } = p - q t\).
2. Plot a scatter diagram of \(\frac { h } { t }\) against \(t\). Given that \(\sum h = 1342 , \sum \frac { h } { t } = 371\) and \(\sum \frac { h ^ { 2 } } { t ^ { 2 } } = 20385\),
3. find the equation of the regression line of \(\frac { h } { t }\) on \(t\) and hence write down the values of \(p\) and \(q\).
4. Use your equation to find the value of \(h\) when \(t = 10\). Comment on the implication of your answer.
5. Find the product-moment correlation coefficient between \(\frac { h } { t }\) and \(t\) and state the significance of its value.
   (4 marks)

7. A new vaccine is tested over a six-month period in one health authority. The table shows the number of new cases of the disease, \(d\), reported in the \(m\) th month after the trials began. A doctor suggests that a relationship of the form \(d = a + b x\) where \(x = \frac { 1 } { m }\) can be used to model the situation.

1. Tabulate the values of \(x\) corresponding to the given values of \(d\) and plot a scatter diagram of \(d\) against \(x\).
2. Explain how your scatter diagram supports the suggested model. You may use $$\Sigma x = 2.45 , \quad \Sigma d = 390 , \quad \Sigma x ^ { 2 } = 1.491 , \quad \Sigma x d = 189.733$$
3. Find an equation of the regression line \(d\) on \(x\) in the form \(d = a + b x\).
4. Use your regression line to estimate how many new cases of the disease there will be in the 13th month after the trial began.
5. Comment on the reliability of your answer to part (d).

1. 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.

The first 14 points are plotted on the axes on page 13.

1. On the axes on page 13, complete the scatter diagram for these data.
2. Use your calculator to write down the equation of the regression line of \(w\) on \(d\).
3. 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).
4. 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 $$\mathrm { S } _ { w u } = 5721.625 \quad \mathrm {~S} _ { u u } = 1482.619 \quad \sum u = 157.57$$
5. By considering the physical nature of the problem, give a reason to support Robert's suggestion.
6. Find the equation of the regression line of \(w\) on \(u\).
7. Find the residual sum of squares (RSS) for Robert's model.
8. State, giving a reason based on these calculations, which of these models better describes these data.
   1. Hence estimate the weight at which a piece of wire with diameter 3 mm will fail. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Question 4 continued} \includegraphics[alt={},max width=\textwidth]{fbd7b196-5372-4956-8d38-92f05c92a5f7-13_2315_1363_301_358}
      \end{figure}

5 A doctors' surgery starts a campaign to reduce missed appointments.
The number of missed appointments for each of the first five weeks after the start of the campaign is shown below. This data could be modelled by an equation of the form \(y = p q ^ { x }\) where \(p\) and \(q\) are constants.

1. Show that this relationship may be expressed in the form \(\log _ { 10 } y = m x + c\), expressing \(m\) and \(c\) in terms of \(p\) and/or \(q\). The diagram below shows \(\log _ { 10 } y\) plotted against \(x\), for the given data.
   \includegraphics[max width=\textwidth, alt={}, center]{54bdddcb-c1ef-4b60-af6c-ac944cae29fe-05_737_1668_1233_258}
2. Estimate the values of \(p\) and \(q\).
3. Use the model to predict when the number of missed appointments will fall below 20. Explain why this answer may not be reliable.
4. A student suggests that, for any prime number between 20 and 40, when its digits are squared and then added, the sum is an odd number. For example, 23 has digits 2 and 3 which gives \(2 ^ { 2 } + 3 ^ { 2 } = 13\), which is odd. Show by counter example that this suggestion is false.
5. Prove that the sum of the squares of any three consecutive positive integers cannot be divided by 3 .