5.09c Calculate regression line

9 Five observations of bivariate data produce the following results, denoted as ( \(x _ { i } , y _ { i }\) ) for \(i = 1,2,3,4,5\). $$\begin{aligned} & ( 13,2.7 ) \\ & { \left[ \Sigma x = 90 , \Sigma y = 15.0 , \Sigma x ^ { 2 } = 1720 , \Sigma y ^ { 2 } = 46.86 , \Sigma x y = 264.0 . \right] } \end{aligned}$$

1. Show that the regression line of \(y\) on \(x\) has gradient - 0.06 , and find its equation in the form \(y = a + b x\).
2. The regression line is used to estimate the value of \(y\) corresponding to \(x = 20\), but the value \(x = 20\) is accurate only to the nearest whole number. Calculate the difference between the largest and the smallest values that the estimated value of \(y\) could take. The numbers \(e _ { 1 } , e _ { 2 } , e _ { 3 } , e _ { 4 } , e _ { 5 }\) are defined by $$e _ { i } = a + b x _ { i } - y _ { i } \quad \text { for } i = 1,2,3,4,5$$
3. The values of \(e _ { 1 } , e _ { 2 }\) and \(e _ { 3 }\) are \(0.6 , - 0.7\) and 0.2 respectively. Calculate the values of \(e _ { 4 }\) and \(e _ { 5 }\).
4. Calculate the value of \(e _ { 1 } ^ { 2 } + e _ { 2 } ^ { 2 } + e _ { 3 } ^ { 2 } + e _ { 4 } ^ { 2 } + e _ { 5 } ^ { 2 }\) and explain the relevance of this quantity to the regression line found in part (i).
5. Find the mean and the variance of \(e _ { 1 } , e _ { 2 } , e _ { 3 } , e _ { 4 } , e _ { 5 }\).

5 A chemical solution was gradually heated. At five-minute intervals the time, \(x\) minutes, and the temperature, \(y ^ { \circ } \mathrm { C }\), were noted. $$\left[ n = 8 , \Sigma x = 140 , \Sigma y = 106.8 , \Sigma x ^ { 2 } = 3500 , \Sigma y ^ { 2 } = 2062.66 , \Sigma x y = 2685.0 . \right]$$

1. Calculate the equation of the regression line of \(y\) on \(x\).
2. Use your equation to estimate the temperature after 12 minutes.
3. It is given that the value of the product moment correlation coefficient is close to + 1 . Comment on the reliability of using your equation to estimate \(y\) when
   1. \(x = 17\),
   2. \(x = 57\).

9 It is thought that the pH value of sand (a measure of the sand's acidity) may affect the extent to which a particular species of plant will grow in that sand. A botanist wished to determine whether there was any correlation between the pH value of the sand on certain sand dunes, and the amount of each of two plant species growing there. She chose random sections of equal area on each of eight sand dunes and measured the pH values. She then measured the area within each section that was covered by each of the two species. The results were as follows. The results for species \(P\) can be summarised by $$n = 8 , \quad \Sigma x = 66.5 , \quad \Sigma x ^ { 2 } = 558.75 , \quad \Sigma y = 1935 , \quad \Sigma y ^ { 2 } = 711275 , \quad \Sigma x y = 17082.5 .$$

1. Give a reason why it might be appropriate to calculate the equation of the regression line of \(y\) on \(x\) rather than \(x\) on \(y\) in this situation.
2. Calculate the equation of the regression line of \(y\) on \(x\) for species \(P\), in the form \(y = a + b x\), giving the values of \(a\) and \(b\) correct to 3 significant figures.
3. Estimate the value of \(y\) for species \(P\) on sand where the pH value is 7.0 . The values of the product moment correlation coefficient between \(x\) and \(y\) for species \(P\) and \(Q\) are \(r _ { P } = 0.828\) and \(r _ { Q } = 0.0302\).
4. Describe the relationship between the area covered by species \(Q\) and the pH value.
5. State, with a reason, whether the regression line of \(y\) on \(x\) for species \(P\) will provide a reliable estimate of the value of \(y\) when the pH value is
   1. 8,
   2. 4 .
   3. Assume that the equation of the regression line of \(y\) on \(x\) for species \(Q\) is also known. State, with a reason, whether this line will provide a reliable estimate of the value of \(y\) when the pH value is 8 .

4 The table shows the latitude, \(x\) (in degrees correct to 3 significant figures), and the average rainfall \(y\) (in cm correct to 3 significant figures) of five European cities. $$\left[ n = 5 , \Sigma x = 265.0 , \Sigma y = 274.6 , \Sigma x ^ { 2 } = 14176.54 , \Sigma y ^ { 2 } = 15162.22 , \Sigma x y = 14464.10 . \right]$$

1. Calculate the product moment correlation coefficient.
2. The values of \(y\) in the table were in fact obtained from measurements in inches and converted into centimetres by multiplying by 2.54 . State what effect it would have had on the value of the product moment correlation coefficient if it had been calculated using inches instead of centimetres.
3. It is required to estimate the annual rainfall at Bergen, where \(x = 60.4\). Calculate the equation of an appropriate line of regression, giving your answer in simplified form, and use it to find the required estimate.

1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows. $$\begin{array} { c c } y \text { on } x : & y = - 0.6 x + 13.0 \\ x \text { on } y : & x = - 1.6 y + 21.0 \end{array}$$

1. State, with a reason, whether the correlation between \(x\) and \(y\) is negative or positive.
2. Neither variable is controlled. Calculate an estimate of the value of \(x\) when \(y = 7.0\).
3. Find the values of \(\bar { x }\) and \(\bar { y }\).

6 A machine with artificial intelligence is designed to improve its efficiency rating with practice. The table shows the values of the efficiency rating, y , after the machine has carried out its task various numbers of times, \(x\) $$\left[ n = 8 , \Sigma x = 60 , \Sigma y = 72 , \Sigma x ^ { 2 } = 1148 , \Sigma y ^ { 2 } = 810 , \Sigma x y = 767 . \right]$$ These data are illustrated in the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{dfad6626-75ca-4dbd-9c45-42f809c163f3-4_769_1328_760_411}

1. (a) Calculate the value of r , the product moment correlation coefficient.
   (b) Without calculation, state with a reason the value of \(\mathrm { r } _ { \mathrm { s } ^ { \prime } }\) Spearman's rank correlation coefficient.
2. A researcher suggests that the data for \(\mathrm { x } = 0\) and \(\mathrm { x } = 1\) should be ignored. Without cal culation, state with a reason what effect this would have on the value of
   (a) \(r\),
   (b) \(r _ { s }\).
3. Use the diagram to estimate the value of y when \(\mathrm { x } = 29\).
4. Jack finds the equation of the regression line of y on xf for all the data, and uses it to estimate the value of \(y\) when \(x = 29\). Without calculation, state with a reason whether this estimate or the one found in part (iii) will be the more reliable.

2

1. The table shows the amount, \(x\), in hundreds of pounds, spent on heating and the number of absences, \(y\), at a factory during each month in 2014.

   Amount, \(x\), spent on heating (£ hundreds)	21	23	19	15	14	5	2	10	9	20	18	23
   Number of absences, \(y\)	23	25	18	18	12	10	4	9	11	15	20	26

   \(n = 12 \quad \Sigma x = 179 \quad \Sigma x ^ { 2 } = 3215 \quad \Sigma y = 191 \quad \Sigma y ^ { 2 } = 3565 \quad \Sigma x y = 3343\)
   1. Calculate \(r\), the product moment correlation coefficient, showing that \(r > 0.92\).
   2. A manager says, 'The value of \(r\) shows that spending more money on heating causes more absences, so we should spend less on heating.' Comment on this claim.
   3. The months in 2014 were numbered \(1,2,3 , \ldots , 12\). The output, \(z\), in suitable units was recorded along with the month number, \(n\), for each month in 2014. The equation of the regression line of \(z\) on \(n\) was found to be \(z = 0.6 n + 17\).
      (a) Use this equation to explain whether output generally increased or decreased over these months.
      (b) Find the mean of \(n\) and use the equation of the regression line to calculate the mean of \(z\).
   4. Hence calculate the total output in 2014.

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. $$\left[ \Sigma x = 43.3 , \Sigma y = 471.9 , \Sigma x ^ { 2 } = 164.69 , \Sigma y ^ { 2 } = 20915.75 , \Sigma x y = 1837.78 . \right]$$ The scatter diagram below illustrates the data. \includegraphics[max width=\textwidth, alt={}, center]{2fb25fc5-0445-44fa-a23e-647d14b1a376-4_462_793_1464_644}

1. Calculate the product moment correlation coefficient for the data, and comment briefly on your answer with reference to the appearance of the scatter diagram.
2. Calculate the equation of the regression line of maximum tidal current on tidal range.
3. 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.
4. 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.

1 A biology student is carrying out an experiment to study the effect of a hormone on the growth of plant shoots. The student applies the hormone at various concentrations to a random sample of twelve shoots and measures the growth of each shoot. The data are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables \(x\) and \(y\), measured in suitable units, represent concentration and growth respectively. \includegraphics[max width=\textwidth, alt={}, center]{20fc4222-95c6-4b59-8e89-913dd988eb44-2_693_897_534_625} $$n = 12 , \Sigma x = 30 , \Sigma y = 967.6 , \Sigma x ^ { 2 } = 90 , \Sigma y ^ { 2 } = 78926 , \Sigma x y = 2530.3 .$$

1. State which of the two variables \(x\) and \(y\) is the independent variable and which is the dependent variable. Briefly explain your answers.
2. Calculate the equation of the regression line of \(y\) on \(x\).
3. Use the equation of the regression line to calculate estimates of shoot growth for concentrations of
   (A) 1.2,
   (B) 4.3. Comment on the reliability of each of these estimates.
4. Calculate the value of the residual for the data point where \(x = 3\) and \(y = 80\).
5. In further experiments, the student finds that using concentration \(x = 6\) results in shoot growths of around \(y = 20\). In the light of all the available information, what can be said about the relationship between \(x\) and \(y\) ?

3 In a triathlon, competitors have to swim 600 metres, cycle 40 kilometres and run 10 kilometres. To improve her strength, a triathlete undertakes a training programme in which she carries weights in a rucksack whilst running. She runs a specific course and notes the total time taken for each run. Her coach is investigating the relationship between time taken and weight carried. The times taken with eight different weights are illustrated on the scatter diagram below, together with the summary statistics for these data. The variables \(x\) and \(y\) represent weight carried in kilograms and time taken in minutes respectively. \includegraphics[max width=\textwidth, alt={}, center]{be463718-caf7-4bc8-b838-143ab4681d6e-4_627_1536_630_281} Summary statistics: \(n = 8 , \Sigma x = 36 , \Sigma y = 214.8 , \Sigma x ^ { 2 } = 204 , \Sigma y ^ { 2 } = 5775.28 , \Sigma x y = 983.6\).

1. Calculate the equation of the regression line of \(y\) on \(x\). On one of the eight runs, the triathlete was carrying 4 kilograms and took 27.5 minutes. On this run she was delayed when she tripped and fell over.
2. Calculate the value of the residual for this weight.
3. The coach decides to recalculate the equation of the regression line without the data for this run. Would it be preferable to use this recalculated equation or the equation found in part (i) to estimate the delay when the triathlete tripped and fell over? Explain your answer. The triathlete's coach claims that there is positive correlation between cycling and swimming times in triathlons. The product moment correlation coefficient of the times of twenty randomly selected competitors in these two sections is 0.209 .
4. Carry out a hypothesis test at the \(5 \%\) level to examine the coach's claim, explaining your conclusions clearly.
5. What distributional assumption is necessary for this test to be valid? How can you use a scatter diagram to decide whether this assumption is likely to be true?

3. A publisher collects information about the amount spent on advertising, \(\pounds x\), and the sales, \(y\) books, for some of her publications. She collects information for a random sample of 8 textbooks and codes the data using \(v = \frac { x + 50 } { 200 }\) and \(s = \frac { y } { 1000 }\) to give [You may use: \(\sum v = 29 \sum s = 36.8 \sum s ^ { 2 } = 209.72 \sum v s = 177.311 \quad \mathrm {~S} _ { v v } = 55.275\) ]

1. Find \(\mathrm { S } _ { v s }\) and \(\mathrm { S } _ { s s }\)
2. Calculate the product moment correlation coefficient for these data. The publisher believes that a linear regression model may be appropriate to describe these data.
3. State, giving a reason, whether or not your answer to part (b) supports the publisher's belief.
4. Find the equation of the regression line of \(s\) on \(v\), giving your answer in the form \(s = a + b v\)
5. Hence find the equation of the regression line of \(y\) on \(x\) for the sample of textbooks, giving your answer in the form \(y = c + d x\) The publisher calculated the regression line for a sample of novels and obtained the equation $$y = 3100 + 1.2 x$$ She wants to increase the sales of books by spending more money on advertising.
6. State, giving your reasons, whether the publisher should spend more money on advertising textbooks or novels.

1. 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.

$$\begin{gathered} \sum x = 504 \quad \sum y = - 27 \quad \sum x ^ { 2 } = 22842 \quad \sum y ^ { 2 } = 62.98 \\ \sum x y = - 1190.7 \quad \mathrm {~S} _ { x x } = 1674 \quad \mathrm {~S} _ { y y } = 2.23 \end{gathered}$$

1. 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.
2. 1. Calculate the product moment correlation coefficient between \(x\) and \(y\)
   2. State, with a reason, whether or not this value supports Priya's belief.
3. 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.
4. 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 \({ } ^ { \circ } \mathrm { F }\)
5. Find an equation of the least squares regression line of \(w\) on \(x\) in the form \(w = c + d x\)
6. Find
   1. the variance of the recorded temperatures when converted to \({ } ^ { \circ } \mathrm { F }\)
   2. the product moment correlation coefficient between \(w\) and \(x\) \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}

3. Martin is investigating the relationship between a person's daily caffeine consumption, \(c\) milligrams, and the amount of sleep they get, \(h\) hours, per night. He collected this information from 20 people and the results are summarised below. $$\begin{array} { c c } \sum c = 3660 \quad \sum h = 126 \quad \sum c ^ { 2 } = 973228 \\ \sum c h = 20023.4 \quad S _ { c c } = 303448 \quad S _ { c h } = - 3034.6 \end{array}$$ Martin calculates the product moment correlation coefficient for these data and obtains - 0.833

1. Give a reason why this value supports a linear relationship between \(c\) and \(h\) The amount of sleep per night is the response variable.
2. Explain what you understand by the term 'response variable'. Martin says that for each additional 100 mg of caffeine consumed, the expected number of hours of sleep decreases by 1
3. Determine, by calculation, whether or not the data support this statement.
4. Use the data to calculate an estimate for the expected number of hours of sleep per night when no caffeine is consumed.
   

5. Franca is the manager of an accountancy firm. She is investigating the relationship between the salary, \(\pounds x\), and the length of commute, \(y\) minutes, for employees at the firm. She collected this information from 9 randomly selected employees. The salary of each employee was then coded using \(w = \frac { x - 20000 } { 1000 }\) The table shows the values of \(w\) and \(y\) for the 9 employees. (You may use \(\sum w = 81 \quad \sum y = 405 \quad \sum w y = 2490 \quad S _ { w w } = 660 \quad S _ { y y } = 2500\) )

1. Calculate the salary of the employee with \(w = - 2\)
2. Show that, to 3 significant figures, the value of the product moment correlation coefficient between \(w\) and \(y\) is - 0.899
3. State, giving a reason, the value of the product moment correlation coefficient between \(x\) and \(y\) The least squares regression line of \(y\) on \(w\) is \(y = 60.75 - 1.75 w\)
4. Find the equation of the least squares regression line of \(y\) on \(x\) giving your answer in the form \(y = a + b x\)
5. Estimate the length of commute for an employee with a salary of \(\pounds 21000\) Franca uses the regression line to estimate the length of commute for employees with salaries between \(\pounds 25000\) and \(\pounds 40000\)
6. State, giving a reason, whether or not these estimates are reliable.

1. Following some school examinations, Chetna is studying the results of the 16 students in her class. The mark for paper \(1 , x\), and the mark for paper \(2 , y\), for each student are summarised in the following statistics.

$$\bar { x } = 35.75 \quad \bar { y } = 25.75 \quad \sigma _ { x } = 7.79 \quad \sigma _ { y } = 11.91 \quad \sum x y = 15837$$

1. Comment on the differences between the marks of the students on paper 1 and paper 2 Chetna decides to examine these data in more detail and plots the marks for each of the 16 students on the scatter diagram opposite.
2. 1. Explain why the circled point \(( 38,0 )\) is possibly an outlier.
   2. Suggest a possible reason for this result. Chetna decides to omit the data point \(( 38,0 )\) and examine the other 15 students' marks.
3. Find the value of \(\bar { x }\) and the value of \(\bar { y }\) for these 15 students. For these 15 students
4. 1. explain why \(\sum x y\) is still 15837
   2. show that \(\mathrm { S } _ { x y } = 1169.8\) For these 15 students, Chetna calculates \(\mathrm { S } _ { x x } = 965.6\) and \(\mathrm { S } _ { y y } = 1561.7\) correct to 1 decimal place.
5. Calculate the product moment correlation coefficient for these 15 students.
6. Calculate the equation of the line of regression of \(y\) on \(x\) for these 15 students, giving your answer in the form \(y = a + b x\) The product moment correlation coefficient between \(x\) and \(y\) for all 16 students is 0.746
7. Explain how your calculation in part (e) supports Chetna's decision to omit the point \(( 38,0 )\) before calculating the equation of the linear regression line.
   (1)
8. Estimate the mark in the second paper for a student who scored 38 marks in the first paper.
   \includegraphics[max width=\textwidth, alt={}]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-17_1127_1146_301_406}
   \includegraphics[max width=\textwidth, alt={}]{d3f4450d-60eb-49b6-be1b-d2fcfad0451f-20_2630_1828_121_121}

1. A research student is investigating the maximum weight, \(y\) grams, of sugar that will dissolve in 100 grams of water at various temperatures, \(x ^ { \circ } \mathrm { C }\), where \(10 \leqslant x \leqslant 80\)

The research student calculated the regression line of \(y\) on \(x\) and found it to be $$y = 151.2 + 2.72 x$$

1. Give an interpretation of the gradient of the regression line.
2. Use the regression line to estimate the maximum weight of sugar that will dissolve in 100 grams of water when the temperature is \(90 ^ { \circ } \mathrm { C }\).
3. Comment on the reliability of your estimate, giving a reason for your answer. Using the regression line of \(y\) on \(x\) and the following summary statistics $$\sum y = 3119 \quad \sum y ^ { 2 } = 851093 \quad \sum x ^ { 2 } = 24500 \quad n = 12$$
4. show that the product moment correlation coefficient for these data is 0.988 to 3 decimal places. The research student's supervisor plotted the original data on a scatter diagram, shown on page 23 With reference to both the scatter diagram and the correlation coefficient,
5. discuss the suitability of a linear regression model to describe the relationship between \(x\) and \(y\).
   \includegraphics[max width=\textwidth, alt={}]{c316fa29-dedc-4890-bd82-31eb0bb819f9-23_990_1138_205_356}

1. A French test and a Spanish test were sat by 11 students.

The table below shows their marks. Greg says that if these points were plotted on a scatter diagram, then the point \(( 30,90 )\) would be an outlier because 90 is an outlier for the Spanish marks. An outlier is defined as a value that is $$\text { greater than } Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { or smaller than } Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)$$

1. Show that 90 is an outlier for the Spanish marks. Ignoring the point (30, 90), Greg calculated the following summary statistics. $$\sum f = 422 \quad \sum s = 382 \quad S _ { f f } = 1667.6 \quad S _ { f s } = 1735.6$$
2. Use these summary statistics to show that the equation of the least squares regression line of \(s\) on \(f\) for the remaining 10 students is $$s = - 5.72 + 1.04 f$$ where the values of the intercept and gradient are given to 3 significant figures. You must show your working.
3. Give an interpretation of the gradient of the regression line. Two further students sat the French test but missed the Spanish test.
4. Using the equation given in part (b), estimate
   1. a Spanish mark for the student who scored 55 marks in their French test,
   2. a Spanish mark for the student who scored 18 marks in their French test.
5. State, giving a reason, which of the two estimates found in part (d) would be the more reliable estimate.

1. A medical researcher is studying the relationship between age ( \(x\) years) and volume of blood ( \(y \mathrm { ml }\) ) pumped by each contraction of the heart. The researcher obtained the following data from a random sample of 8 patients.

[You may use \(\sum x = 370 , \mathrm {~S} _ { x x } = 2587.5 , \sum y = 560 , \sum y ^ { 2 } = 39418 , \mathrm {~S} _ { x y } = - 710\) ]

1. Calculate \(\mathrm { S } _ { y y }\)
2. Calculate the product moment correlation coefficient for these data.
3. Interpret your value of the correlation coefficient. The researcher believes that a linear regression model may be appropriate to describe these data.
4. State, giving a reason, whether or not your value of the correlation coefficient supports the researcher's belief.
5. Find the equation of the regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\) Jack is a 40-year-old patient.
6. 1. Use your regression line to estimate the volume of blood pumped by each contraction of Jack's heart.
   2. Comment, giving a reason, on the reliability of your estimate.

2. Paul believes there is a relationship between the value and the floor size of a house. He takes a random sample of 20 houses and records the value, \(\pounds v\), and the floor size, \(s \mathrm {~m} ^ { 2 }\) The data were coded using \(x = \frac { s - 50 } { 10 }\) and \(y = \frac { v } { 100000 }\) and the following statistics obtained. $$\sum x = 441.5 , \quad \sum y = 59.8 , \quad \sum x ^ { 2 } = 11261.25 , \quad \sum y ^ { 2 } = 196.66 , \quad \sum x y = 1474.1$$

1. Find the value of \(S _ { x y }\) and the value of \(S _ { x x }\)
2. Find the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\) The least squares regression line of \(v\) on \(s\) is \(v = c + d s\)
3. Show that \(d = 1020\) to 3 significant figures and find the value of \(c\)
4. Estimate the value of a house of floor size \(130 \mathrm {~m} ^ { 2 }\)
5. Interpret the value \(d\) Paul wants to increase the value of his house. He decides to add an extension to increase the floor size by \(31 \mathrm {~m} ^ { 2 }\)
6. Estimate the increase in the value of Paul's house after adding the extension.

1. An office has the heating switched on at 7.00 a.m. each morning. On a particular day, the temperature of the office, \(t { } ^ { \circ } \mathrm { C }\), was recorded \(m\) minutes after 7.00 a.m. The results are shown in the table below.

1. Calculate the exact values of \(S _ { m t }\) and \(S _ { m m }\).
2. Calculate the equation of the regression line of \(t\) on \(m\) in the form \(t = a + b m\).
3. Use your equation to estimate the value of \(t\) at 7.35 a.m.
4. State, giving a reason, whether or not you would use the regression equation in (b) to estimate the temperature
   1. at 9.00 a.m. that day,
   2. at 7.15 a.m. one month later.

2 The table shows the age, \(x\) years, and the mean diameter, \(y \mathrm {~cm}\), of the trunk of each of seven randomly selected trees of a certain species. $$\left[ n = 7 , \Sigma x = 202 , \Sigma y = 245.3 , \Sigma x ^ { 2 } = 7300 , \Sigma y ^ { 2 } = 10510.65 , \Sigma x y = 8736.9 . \right]$$

1. (a) Use an appropriate formula to show that the gradient of the regression line of \(y\) on \(x\) is 1.13 , correct to 2 decimal places.
   (b) Find the equation of the regression line of \(y\) on \(x\).
2. Use your equation to estimate the mean trunk diameter of a tree of this species with age
   (a) 30 years,
   (b) 100 years. It is given that the value of the product moment correlation coefficient for the data in the table is 0.988 , correct to 3 decimal places.
3. Comment on the reliability of each of your two estimates.

3 A firm wishes to assess whether there is a linear relationship between the annual amount spent on advertising, \(\pounds x\) thousand, and the annual profit, \(\pounds y\) thousand. A summary of the figures for 12 years is as follows. $$n = 12 \quad \Sigma x = 86.6 \quad \Sigma y = 943.8 \quad \Sigma x ^ { 2 } = 658.76 \quad \Sigma y ^ { 2 } = 83663.00 \quad \Sigma x y = 7351.12$$

1. Calculate the product moment correlation coefficient, showing that it is greater than 0.9 .
2. Comment briefly on this value in this context.
3. A manager claims that this result shows that spending more money on advertising in the future will result in greater profits. Make two criticisms of this claim.
4. Calculate the equation of the regression line of \(y\) on \(x\).
5. Estimate the annual profit during a year when \(\pounds 7400\) was spent on advertising.

2 In an experiment, the percentage sand content, \(y\), of soil in a given region was measured at nine different depths, \(x \mathrm {~cm}\), taken at intervals of 6 cm from 0 cm to 48 cm . The results are summarised below. $$n = 9 \quad \Sigma x = 216 \quad \Sigma x ^ { 2 } = 7344 \quad \Sigma y = 512.4 \quad \Sigma y ^ { 2 } = 30595 \quad \Sigma x y = 10674$$

1. State, with a reason, which variable is the independent variable.
2. Calculate the product moment correlation coefficient between \(x\) and \(y\).
3. (a) Calculate the equation of the appropriate regression line.
   (b) This regression line is used to estimate the percentage sand content at depths of 25 cm and 100 cm . Comment on the reliability of each of these estimates. You are not asked to find the estimates.

1 For each of the last five years the number of tourists, \(x\) thousands, visiting Sackton, and the average weekly sales, \(\pounds y\) thousands, in Sackton Stores were noted. The table shows the results.

1. Calculate the product moment correlation coefficient \(r\) between \(x\) and \(y\).
2. It is required to estimate the average weekly sales at Sackton Stores in a year when the number of tourists is 280000 . Calculate the equation of an appropriate regression line, and use it to find this estimate.
3. Over a longer period the value of \(r\) is - 0.8 . The mayor says, "This shows that having more tourists causes sales at Sackton Stores to decrease." Give a reason why this statement is not correct.