2.02c Scatter diagrams and regression lines

6 [Figure 1, printed on the insert, is provided for use in this question.]
For a random sample of 10 patients who underwent hip-replacement operations, records were kept of their ages, \(x\) years, and of the number of days, \(y\), following their operations before they were able to walk unaided safely.

1. On Figure 1, complete the scatter diagram for these data.
2. Calculate the equation of the least squares regression line of \(y\) on \(x\).
3. Draw your regression line on Figure 1.
4. In fact, patients H, I and J were males and the other 7 patients were females.
   1. Calculate the mean of the residuals for the 3 male patients.
   2. Hence estimate, for a male patient aged 65 years, the number of days following his hip-replacement operation before he is able to walk unaided safely.

2 Hermione, who is studying reptiles, measures the length, \(x \mathrm {~cm}\), and the weight, \(y\) grams, of a sample of 11 adult snakes of the same type. Her results are shown in the table.

3 The table shows the maximum weight, \(y _ { A }\) grams, of Salt \(A\) that will dissolve in 100 grams of water at various temperatures, \(x ^ { \circ } \mathrm { C }\).

1. Calculate the equation of the least squares regression line of \(y _ { A }\) on \(x\).
2. The data in the above table are plotted on the scatter diagram on page 4. Draw your regression line on this scatter diagram.
3. For water temperatures in the range \(10 ^ { \circ } \mathrm { C }\) to \(80 ^ { \circ } \mathrm { C }\), the maximum weight, \(y _ { B }\) grams, of Salt \(B\) that will dissolve in 100 grams of water is given by the equation $$y _ { B } = 60.1 + 0.255 x$$
   1. Draw this line on the scatter diagram.
   2. Estimate the water temperature at which the maximum weight of Salt \(A\) that will dissolve in 100 grams of water is the same as that of Salt B.
   3. For Salt \(A\) and Salt \(B\), compare the effects of water temperature on the maximum weight that will dissolve in 100 grams of water. Your answer should identify two distinct differences. \section*{Temperatures and Maximum Weights}
      \includegraphics[max width=\textwidth, alt={}]{91466019-8feb-4292-b616-e8e8667e2e54-4_2023_1682_404_173}

5 As part of a study of charity shops in a small market town, two such shops, \(X\) and \(Y\), were each asked to provide details of its takings on 12 randomly selected days. The table shows, for each of the 12 days, the day's takings, \(\pounds x\), of charity shop \(X\) and the day's takings, \(\pounds y\), of charity shop \(Y\).

1. 1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
   2. Interpret your value in the context of this question.
2. Complete the scatter diagram shown on the opposite page.
3. The investigator realised subsequently that one of the 12 selected days was a particularly popular town market day and another was a day on which the weather was extremely severe. Identify each of these days giving a reason for each choice.
4. Removing the two days described in part (c) from the data gives the following information. $$S _ { x x } = 1292.5 \quad S _ { y y } = 3850.1 \quad S _ { x y } = 407.5$$
   1. Use this information to recalculate the value of the product moment correlation coefficient between \(x\) and \(y\).
   2. Hence revise, as necessary, your interpretation in part (a)(ii).
      [0pt] [3 marks] Shop \(X\) takings(£) \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{harity Shops} \includegraphics[alt={},max width=\textwidth]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_33_21_294_1617}
      \end{figure} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{harity Shops} \includegraphics[alt={},max width=\textwidth]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_49_24_276_1710}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{ddf7f158-b6ae-42c6-98f1-d59c205646ad-17_1304_415_406_1391}
      レ

1 The table shows the heights, \(x \mathrm {~cm}\), and the arm spans, \(y \mathrm {~cm}\), of a random sample of 12 men aged between 21 years and 40 years.

1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
2. Interpret, in context, your value calculated in part (a).

4 As part of her science project, a student found the mass, \(y\) grams, of a particular compound that dissolved in 100 ml of water at each of 12 different set temperatures, \(x ^ { \circ } \mathrm { C }\). The results are shown in the table.

1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
2. Interpret, in context, your value for the gradient of this regression line.
3. Use your equation to estimate the mass of the compound which will dissolve in 100 ml of water at \(68 ^ { \circ } \mathrm { C }\).
4. Given that the values of the 12 residuals for the regression line of \(y\) on \(x\) lie between - 7 and + 9 , comment, with justification, on the likely accuracy of your estimate in part (c).
   [0pt] [2 marks]

1. The table shows the numbers of cars and vans in a company's fleet having registrations with the prefix letters shown.

1. Plot a scatter graph of this data, with the number of cars on the horizontal axis and the number of vans on the vertical axis.
2. If there were \(4 J\)-registered cars, estimate the number of \(J\)-registered vans. Given that \(\sum x ^ { 2 } = 1001 , \sum y ^ { 2 } = 1264\) and \(\sum x y = 1106\),
3. calculate the product-moment correlation coefficient between \(x\) and \(y\). Give a brief interpretation of your answer.

7. A doctor wished to investigate the effects of staying awake for long periods on a person's ability to complete simple tasks. She recorded the number of times, \(n\), that a subject could clinch his or her fist in 30 seconds after being awake for \(h\) hours. The results for one subject were as follows.

1. Plot a scatter diagram of \(n\) against \(h\) for these results. You may use $$\Sigma h = 180 , \quad \Sigma n = 875 , \quad \Sigma h ^ { 2 } = 3660 , \quad \Sigma h n = 17204 .$$
2. Obtain the equation of the regression line of \(n\) on \(h\) in the form \(n = a + b h\).
3. Give a practical interpretation of the constant b.
4. Explain why this regression line would be unlikely to be appropriate for values of \(h\) between 0 and 16 .
   (2 marks)
   Another subject underwent the same tests giving rise to a regression line of \(n = 213.4 - 5.87\) h
5. After how many hours of being awake together would you expect these two subjects to be able to clench their fists the same number of times in 30 seconds?

6. A school introduced a new programme of support lessons in 1994 with a view to improving grades in GCSE English. The table below shows the number of years since 1994, n, and the corresponding percentage of students achieving A to C grades in GCSE English, \(p\), for each year.

1. Represent these data on a scatter diagram. You may use the following values. $$\Sigma n = 21 , \quad \Sigma p = 240.1 , \quad \Sigma n ^ { 2 } = 91 , \quad \Sigma p ^ { 2 } = 9675.41 , \quad \Sigma n p = 873 .$$
2. Find an equation of the regression line of \(p\) on \(n\) and draw it on your graph.
3. Calculate the product moment correlation coefficient for these data and comment on the suitability of a linear model for the relationship between \(n\) and \(p\) during this period.

7. Pipes-R-us manufacture a special lightweight aluminium tubing. The price \(\pounds P\), for each length, \(l\) metres, that the company sells is shown in the table.

1. Represent these data on a scatter diagram. You may use $$\Sigma l = 15.8 , \quad \Sigma P = 46.6 , \quad \Sigma l ^ { 2 } = 60.14 , \quad \Sigma l P = 159.77$$
2. Find the equation of the regression line of \(P\) on \(l\) in the form \(P = a + b l\).
3. Give a practical interpretation of the constant b. In response to customer demand Pipes- \(R\)-us decide to start selling tubes cut to specific lengths. Initially the company decides to use the regression line found in part (b) as a pricing formula for this new service.
4. Calculate the price that Pipes- \(R\)-us should charge for 5.2 metres of the tubing.
5. Suggest a reason why Pipes- \(R\)-us might not offer prices based on the regression line for any length of tubing.

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).

6. A physics student recorded the length, \(l \mathrm {~cm}\), of a spring when different masses, \(m\) grams, were suspended from it giving the following results.

1. Represent these data on a scatter diagram with \(l\) on the vertical axis. The student decides to find the equation of a regression line of the form \(l = a + b m\) using only the data for \(m \leq 500 \mathrm {~g}\).
2. Give a reason to support the fitting of such a regression line and explain why the student is excluding two of his values.
   (2 marks)
   You may use $$\Sigma m = 1550 , \quad \Sigma l = 119 , \quad \Sigma m ^ { 2 } = 552500 , \quad \Sigma l ^ { 2 } = 2869.2 , \quad \Sigma m l = 39540 .$$
3. Find the values of \(a\) and \(b\).
4. Explain the significance of the values of \(a\) and \(b\) in this situation.

4 [Figure 1 and Figure 2, printed on the insert, are provided for use in this question.]
The variables \(x\) and \(y\) are related by an equation of the form $$y = a x + \frac { b } { x + 2 }$$ where \(a\) and \(b\) are constants.

1. The variables \(X\) and \(Y\) are defined by \(X = x ( x + 2 ) , Y = y ( x + 2 )\). Show that \(Y = a X + b\).
2. The following approximate values of \(x\) and \(y\) have been found:

   \(x\)	1	2	3	4
   \(y\)	0.40	1.43	2.40	3.35

   1. Complete the table in Figure 1, showing values of \(X\) and \(Y\).
   2. Draw on Figure 2 a linear graph relating \(X\) and \(Y\).
   3. Estimate the values of \(a\) and \(b\).

4 The variables \(x\) and \(y\) are known to be related by an equation of the form $$y = a b ^ { x }$$ where \(a\) and \(b\) are constants.

1. Given that \(Y = \log _ { 10 } y\), show that \(x\) and \(Y\) must satisfy an equation of the form $$Y = m x + c$$
2. The diagram shows the linear graph which has equation \(Y = m x + c\). \includegraphics[max width=\textwidth, alt={}, center]{932d4c7e-6514-4543-b1d1-753fca5a08fd-5_744_720_833_699} Use this graph to calculate:
   1. an approximate value of \(y\) when \(x = 2.3\), giving your answer to one decimal place;
   2. an approximate value of \(x\) when \(y = 80\), giving your answer to one decimal place.
      (You are not required to find the values of \(m\) and \(c\).)

4 The variables \(x\) and \(y\) are related by an equation of the form $$y = a x ^ { 2 } + b$$ where \(a\) and \(b\) are constants.
The following approximate values of \(x\) and \(y\) have been found.

1. Complete the table below, showing values of \(X\), where \(X = x ^ { 2 }\).
2. On the diagram below, draw a linear graph relating \(X\) and \(y\).
3. Use your graph to find estimates, to two significant figures, for:
   1. the value of \(x\) when \(y = 15\);
   2. the values of \(a\) and \(b\).

      1. \(\boldsymbol { x }\)	2	4	6	8
         \(\boldsymbol { X }\)
         \(\boldsymbol { y }\)	6.0	10.5	18.0	28.2

      2. \includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-05_771_1586_1772_274}

2 A researcher is investigating the concentration of bacteria and fungi in the air in buildings. The researcher selects a random sample of 12 buildings and measures the concentrations of bacteria, \(x\), and fungi, \(y\), in the air in each building. Both concentrations are measured in the same standard units. Fig. 2 illustrates the data collected. The researcher wishes to test for a relationship between \(x\) and \(y\). \begin{figure}[h] \end{figure}

1. Explain why a test based on the product moment correlation coefficient is likely to be appropriate for these data. Summary statistics for the data are as follows. \(n = 12 \quad \sum x = 18030 \quad \sum y = 15550 \quad \sum x ^ { 2 } = 31458700 \quad \sum y ^ { 2 } = 21980500 \quad \sum x y = 25626800\)
2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient between \(x\) and \(y\).
3. Carry out a test at the \(5 \%\) significance level based on the product moment correlation coefficient to investigate whether there is any correlation between concentrations of bacteria and fungi.
4. Explain why, in order for proper inference to be undertaken, the sample should be chosen randomly.

3 A student is investigating the link between temperature (in degrees Celsius) and electricity consumption (in Gigawatt-hours) in the country in which he lives. The student has read that there is strong negative correlation between daily mean temperature over the whole country and daily electricity consumption during a year. He wonders if this applies to an individual season. He therefore obtains data on the mean temperature and electricity consumption on ten randomly selected days in the summer. The spreadsheet output below shows the data, together with a scatter diagram to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{5be067ff-4668-48d6-8ed2-b8dfa3e678f7-3_798_1593_639_251}

1. Calculate Pearson's product moment correlation coefficient between daily mean temperature and daily electricity consumption. The student decides to carry out a hypothesis test to investigate whether there is negative correlation between daily mean temperature and daily electricity consumption during the summer.
2. Explain why the student decides to carry out a test based on Pearson's product moment correlation coefficient.
3. Show that the test at the \(5 \%\) significance level does not result in the null hypothesis being rejected.
4. The student concludes that there is no correlation between the variables in the summer months. Comment on the student's conclusion.

5 A student wants to know if there is a positive correlation between the amounts of two pollutants, sulphur dioxide and PM10 particulates, on different days in the area of London in which he lives; these amounts, measured in suitable units, are denoted by \(s\) and \(p\) respectively.
He uses a government website to obtain data for a random sample of 15 days on which the amounts of these pollutants were measured simultaneously. Fig. 5.1 is a scatter diagram showing the data. Summary statistics for these 15 values of \(s\) and \(p\) are as follows. \(\sum s _ { 1 } = 155.4 \quad \sum p = 518.9 \quad \sum s ^ { 2 } = 2322.7 \quad \sum p ^ { 2 } = 21270.5 \quad \sum s p = 6009.1\) \begin{figure}[h] \end{figure}

1. Explain why the student might come to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.
2. Find the value of Pearson's product moment correlation coefficient.
3. Carry out a test at the \(5 \%\) significance level to investigate whether there is positive correlation between the amounts of sulphur dioxide and PM10 particulates.
4. Explain why the student made sure that the sample chosen was a random sample. The student also wishes to model the relationship between the amounts of nitrogen dioxide \(n\) and PM10 particulates \(p\).
   He takes a random sample of 54 values of the two variables, both measured at the same times. Fig. 5.2 is a scatter diagram which shows the data, together with the regression line of \(n\) on \(p\), the equation of the regression line and the value of \(r ^ { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4a4d5816-5b53-49a1-b72f-f8bcf3b4e8bc-5_824_1230_495_258} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}
5. Predict the value of \(n\) for \(p = 150\).
6. Discuss the reliability of your prediction in part (e).

2 A forester is investigating the relationship between the diameter and the height of young beech trees. She selects a random sample of 15 young beech trees in a forest and records their diameters, \(d \mathrm {~cm}\), and their heights, \(h \mathrm {~m}\). The data are illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{e8624e9b-5143-49d2-9683-cc3a1082694e-3_649_1116_386_230}

1. State whether either or both of the variables \(d\) and \(h\) are random variables. Summary data for the diameters and heights are as follows. $$\mathrm { n } = 15 \quad \sum \mathrm {~d} = 84.9 \quad \sum \mathrm {~h} = 124.7 \quad \sum \mathrm {~d} ^ { 2 } = 624.55 \quad \sum \mathrm {~h} ^ { 2 } = 1230.57 \quad \sum \mathrm { dh } = 866.63$$
2. Find the equation of the regression line of \(h\) on \(d\). Give your answer in the form \(h = a d + b\), giving the values of \(a\) and \(b\) correct to \(\mathbf { 2 }\) decimal places.
3. Use the regression line to predict the heights of beech trees with the following diameters.
   Comment on this in relation to your regression line.
4. State the coordinates of the point at which the regression line of \(d\) on \(h\) meets the line which you calculated in part (b).

4 A scientist is investigating sea salinity (the level of salt in the sea) in a particular area. She wishes to check whether satellite measurements, \(y\), of salinity are similar to those directly measured, \(x\). Both variables are measured in parts per thousand in suitable units. The scientist obtains a random sample of 10 values of \(x\) and the related values of \(y\). Below is a screenshot of a scatter diagram to illustrate the data. She decides to carry out a hypothesis test to check if there is any correlation between direct measurement, \(x\), and satellite measurement, \(y\). \includegraphics[max width=\textwidth, alt={}, center]{691e8b55-e9a1-4fff-b9ee-a71ff1f73ead-5_830_837_589_246}

1. Explain why the scientist might decide to carry out a test based on the product moment correlation coefficient. Summary statistics for \(x\) and \(y\) are as follows. \(n = 10 \quad \sum x = 351.9 \quad \sum y = 350.0 \quad \sum x ^ { 2 } = 12384.5 \quad \sum y ^ { 2 } = 12251.2 \quad \sum \mathrm { xy } = 12317.2\)
2. In this question you must show detailed reasoning. Calculate the product moment correlation coefficient.
3. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is positive correlation between directly measured and satellite measured salinity levels.
4. Explain why it would be preferable to use a larger sample. The scientist is also interested in whether there is any correlation between salinity and numbers of a particular species of shrimp in the water. She takes a large sample and finds that the product moment correlation coefficient for this sample is 0.165 . The result of a test based on this sample is to reject the null hypothesis and conclude that there is correlation between salinity and numbers of shrimp.
5. Comment on the outcome of the hypothesis test with reference to the effect size of 0.165 .

8 A swimming coach is investigating whether there is correlation between the times taken by teenage swimmers to swim 50 m Butterfly and 50 m Freestyle. The coach selects a random sample of 11 teenage swimmers and records the times that each of them take for each event. The spreadsheet shows the data, together with a scatter diagram to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-06_712_1465_456_274}

1. In the scatter diagram, Butterfly times have been plotted on the horizontal axis and Freestyle times on the vertical axis. A student states that the variables should have been plotted the other way around. Explain whether the student is correct. The student decides to carry out a hypothesis test to investigate whether there is any correlation between the times taken for the two events.
2. Explain why the student decides to carry out a test based on Spearman's rank correlation coefficient.
3. In this question you must show detailed reasoning. Carry out the test at the 5\% significance level.
4. The student concludes that there is definitely no correlation between the times. Comment on the student's conclusion.

13 Answer part (ii) of this question on the insert provided. The table gives a firm's monthly profits for the first few months after the start of its business, rounded to the nearest \(\pounds 100\). The firm's profits, \(\pounds y\), for the \(x\) th month after start-up are modelled by $$y = k \times 10 ^ { a x }$$ where \(a\) and \(k\) are constants.

1. Show that, according to this model, a graph of \(\log _ { 10 } y\) against \(x\) gives a straight line of gradient \(a\) and intercept \(\log _ { 10 } k\).
2. On the insert, complete the table and plot \(\log _ { 10 } y\) against \(x\), drawing by eye a line of best fit.
3. Use your graph to find an equation for \(y\) in terms of \(x\) for this model.
4. For which month after start-up does this model predict profits of about \(\pounds 75000\) ?
5. State one way in which this model is unrealistic.