5.09d Linear coding: effect on regression

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. \(x = 57\).

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.

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.

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.

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 Five salesmen from a certain firm were selected at random for a survey. For each salesman, the annual income, \(x\) thousand pounds, and the distance driven last year, \(y\) thousand miles, were recorded. The results were summarised as follows. $$n = 5 \quad \Sigma x = 251 \quad \Sigma x ^ { 2 } = 14323 \quad \Sigma y = 65 \quad \Sigma y ^ { 2 } = 855 \quad \Sigma x y = 3247$$

1. (a) Show that the product moment correlation coefficient, \(r\), between \(x\) and \(y\) is - 0.122 , correct to 3 significant figures.
   (b) State what this value of \(r\) shows about the relationship between annual income and distance driven last year for these five salesmen.
   (c) It was decided to recalculate \(r\) with the distances measured in kilometres instead of miles. State what effect, if any, this would have on the value of \(r\).
2. Another salesman from the firm is selected at random. His annual income is known to be \(\pounds 52000\), but the distance that he drove last year is unknown. In order to estimate this distance, a regression line based on the above data is used. Comment on the reliability of such an estimate.

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.

1 A random sample of wheat seedlings is planted and their growth is measured. The table shows their average growth, \(y \mathrm {~mm}\), at half-day intervals.

1. Draw a scatter diagram to illustrate these data.
2. Calculate the equation of the regression line of \(y\) on \(t\).
3. Calculate the value of the residual for the data point at which \(t = 2\).
4. Use the equation of the regression line to calculate an estimate of the average growth after 5 days for wheat seedlings. Comment on the reliability of this estimate. It is suggested that it would be better to replace the regression line by a line which passes through the origin. You are given that the equation of such a line is \(y = a t\), where \(a = \frac { \sum y t } { \sum t ^ { 2 } }\).
5. Find the equation of this line and plot the line on your scatter diagram.

10 The values from a random sample of five pairs \(( x , y )\) taken from a bivariate distribution are shown below. The equation of the regression line of \(x\) on \(y\) is given by \(x = \frac { 5 } { 4 } y + c\).

1. Given that \(q\) is an integer, find its value.
2. Find the value of \(c\).
3. Find the value of the product moment correlation coefficient.

The regression line of \(y\) on \(x\) obtained from a random sample of five pairs of values of \(x\) and \(y\) is $$y = 2.5 x - 1.5$$ The data is given in the following table.

1. Show that \(p + q = 19\).
2. Find the values of \(p\) and \(q\).
3. Determine the value of the product moment correlation coefficient for this sample.
4. It is later discovered that the values of \(x\) given in the table have each been divided by 10 (that is, the actual values are \(10,20,40,20,60\) ). Without any further calculation, state
   1. the equation of the actual regression line of \(y\) on \(x\),
   2. the value of the actual product moment correlation coefficient.

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]{d138173d-c70c-46db-b9b9-d5f19334c5f1-04_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?

6. Anna is investigating the relationship between exercise and resting heart rate. She takes a random sample of 19 people in her year at school and records for each person

• their resting heart rate, \(h\) beats per minute
• the number of minutes, \(m\), spent exercising each week

Her results are shown on the scatter diagram. \includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-16_531_551_653_740}

1. Interpret the nature of the relationship between \(h\) and \(m\) Anna codes the data using the formulae $$\begin{aligned} & x = \log _ { 10 } m \\ & y = \log _ { 10 } h \end{aligned}$$ The product moment correlation coefficient between \(x\) and \(y\) is - 0.897
2. Test whether or not there is significant evidence of a negative correlation between \(x\) and \(y\) You should
   The equation of the line of best fit of \(y\) on \(x\) is $$y = - 0.05 x + 1.92$$
3. Use the equation of the line of best fit of \(y\) on \(x\) to find a model for \(h\) on \(m\) in the form $$h = a m ^ { k }$$ where \(a\) and \(k\) are constants to be found.

9 The pre-release material contains information concerning the median income of taxpayers in different areas of London. Some of the data for Camden is shown in the table below. The years quoted in this question refer to the end of the financial years used in the pre-release material. For example, the year 2004 in the table refers to the year 2003/04 in the pre-release material.

1. Explain whether these data are a sample or a population of Camden taxpayers. A time series for the data is shown below. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Median income of taxpayers in Camden 2004-2011} \includegraphics[alt={},max width=\textwidth]{11788aaf-98fb-4a78-8a40-a40743b1fe15-07_624_1469_950_242}
   \end{figure} The LINEST function on a spreadsheet is used to formulate the following model for the data: \(I = 1115 Y - 2212950\), where \(I =\) median income of taxpayers in \(\pounds\) and \(Y =\) year.
2. Use this model to find an estimate of the median income of taxpayers in Camden in 2009.
3. Give two reasons why this estimate is likely to be close to the true value. The median income of taxpayers in Croydon in 2009 is also not available.
4. Use your knowledge of the pre-release material to explain whether the model used in part (b) would give a reasonable estimate of the missing value for Croydon.

3

1. Using the scatter diagram in the Printed Answer Booklet, explain what is meant by least squares in the context of a regression line of \(y\) on \(x\).
2. A set of bivariate data \(( t , u )\) is summarised as follows. \(n = 5 \quad \sum t = 35 \quad \sum u = 54\) \(\sum t ^ { 2 } = 285 \quad \sum u ^ { 2 } = 758 \quad \sum \mathrm { tu } = 460\)
   1. Calculate the equation of the regression line of \(u\) on \(t\).
   2. The variables \(t\) and \(u\) are now scaled using the following scaling. \(\mathrm { v } = 2 \mathrm { t } , \mathrm { w } = \mathrm { u } + 4\) Find the equation of the regression line of \(w\) on \(v\), giving your equation in the form \(w = f ( v )\).

7 The coordinates of a set of 10 points are denoted by ( \(\mathrm { x } _ { \mathrm { i } } , \mathrm { y } _ { \mathrm { i } }\) ) for \(i = 1,2 , \ldots , 10\). For a particular set of values of ( \(\mathrm { x } _ { \mathrm { i } } , \mathrm { y } _ { \mathrm { i } }\) ) and any constants \(a\) and \(b\) it can be shown that \(\Sigma \left( y _ { i } - a - b x _ { i } \right) ^ { 2 } = 10 ( 11 - a - 6 b ) ^ { 2 } + 126 \left( b - \frac { 83 } { 42 } \right) ^ { 2 } + \frac { 139 } { 14 }\).

1. 1. Explain why \(\sum \left( \mathrm { y } _ { \mathrm { i } } - \mathrm { a } - \mathrm { bx } _ { \mathrm { i } } \right) ^ { 2 }\) is minimised by taking \(b = \frac { 83 } { 42 }\) and \(\mathrm { a } = 11 - 6 \mathrm {~b}\).
   2. Hence explain why the equation of the regression line of \(y\) on \(x\) for these points is given by the corresponding values of \(a\) and \(b\) (so that the equation is \(\mathrm { y } = \frac { 83 } { 42 } \mathrm { x } - \frac { 6 } { 7 }\) ).
2. State which of the following terms cannot apply to the variable \(X\) if the regression line of \(y\) on \(x\) can be used for estimating values of \(Y\). Dependent Independent Controlled Response
3. Use the regression line to estimate the value of \(y\) corresponding to \(x = 8\).
4. State what must be true of the value \(x = 8\) if the estimate in part (c) is to be reliable.
5. Variables \(u\) and \(v\) are related to \(x\) and \(y\) by the following relationships. \(u = 2 + 4 x \quad v = 8 - 2 y\) Show that the gradient of the regression line of \(v\) on \(u\) is very close to - 1 .

1 At a seaside resort the number \(X\) of ice-creams sold and the temperature \(Y ^ { \circ } \mathrm { F }\) were recorded on 20 randomly chosen summer days. The data can be summarised as follows. \(\sum x = 1506 \quad \sum x ^ { 2 } = 127542 \quad \sum y = 1431 \quad \sum y ^ { 2 } = 104451 \quad \sum x y = 111297\)

1. Calculate the equation of the least squares regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\).
2. Explain the significance for the regression line of the quantity \(\sum \left[ y _ { i } - \left( a x _ { i } + b \right) \right] ^ { 2 }\).
3. It is decided to measure the temperature in degrees Centigrade instead of degrees Fahrenheit. If the same temperature is measured both as \(f ^ { \circ }\) Fahrenheit and \(c ^ { \circ }\) Centigrade, the relationship between \(f\) and \(c\) is \(\mathrm { c } = \frac { 5 } { 9 } ( \mathrm { f } - 32 )\). Find the equation of the new regression line.

1. Stuart is investigating the relationship between Gross Domestic Product (GDP) and the size of the population for a particular country.
   He takes a random sample of 9 years and records the size of the population, \(t\) millions, and the GDP, \(g\) billion dollars for each of these years.

The data are summarised as $$n = 9 \quad \sum t = 7.87 \quad \sum g = 144.84 \quad \sum g ^ { 2 } = 3624.41 \quad S _ { t t } = 1.29 \quad S _ { t g } = 40.25$$

1. Calculate the product moment correlation coefficient between \(t\) and \(g\)
2. Give an interpretation of your product moment correlation coefficient.
3. Find the equation of the least squares regression line of \(g\) on \(t\) in the form \(g = a + b t\)
4. Give an interpretation of the value of \(b\) in your regression line.
5. 1. Use the regression line from part (c) to estimate the GDP, in billions of dollars, for a population of 7000000
   2. Comment on the reliability of your answer in part (i). Give a reason, in context, for your answer. Using the regression line from part (c), Stuart estimates that for a population increase of \(x\) million there will be an increase of 0.1 billion dollars in GDP.
6. Find the value of \(x\)

1. A biologist is studying bears. The biologist records the length, \(d \mathrm {~cm}\), and the girth, \(g \mathrm {~cm}\), of 8 bears. The biologist summarises the data as follows

$$\begin{gathered} \sum d = 1456.8 \quad \sum g = 713.2 \quad \sum d g = 141978.84 \quad \sum g ^ { 2 } = 72675.98 \\ S _ { d d } = 16769.78 \end{gathered}$$

1. Calculate the exact value of \(S _ { d g }\) and the exact value of \(S _ { g g }\)
2. Calculate the value of the product moment correlation coefficient between \(d\) and \(g\)
3. Show that the equation of the regression line of \(g\) on \(d\) can be written as $$g = - 42.3 + 0.722 d$$ where the values of the intercept and gradient are given to 3 significant figures.
4. Give an interpretation, in context, of the gradient of the regression line. Using the equation of the regression line given in part (c)
5. 1. estimate the girth of a bear with a length of 2.5 metres,
   2. explain why an estimate for the girth of a bear with a length of 0.5 metres is not reliable. Using the regression line from part (c), the biologist estimates that for each \(x \mathrm {~cm}\) increase in the length of a bear there will be a 17.3 cm increase in the girth.
6. Find the value of \(x\)

1. The production cost, \(\pounds c\) million, of a film and the total ticket sales, \(\pounds t\) million, earned by the film are recorded for a sample of 40 films.

Some summary statistics are given below. $$\sum c = 1634 \quad \sum t = 1361 \quad \sum t ^ { 2 } = 82873 \quad \sum c t = 83634 \quad \mathrm {~S} _ { c c } = 28732.1$$

1. Find the exact value of \(\mathrm { S } _ { t t }\) and the exact value of \(\mathrm { S } _ { c t }\)
2. Calculate the value of the product moment correlation coefficient for these data.
3. Give an interpretation of your answer to part (b)
4. Show that the equation of the linear regression line of \(t\) on \(c\) can be written as $$t = - 5.84 + 0.976 c$$ where the values of the intercept and gradient are given to 3 significant figures.
5. Find the expected total ticket sales for a film with a production cost of \(\pounds 90\) million. Using the regression line in part (d)
6. find the range of values of the production cost of a film for which the total ticket sales are less than \(80 \%\) of its production cost.

1. The percentage oil content, \(p\), and the weight, \(w\) milligrams, of each of 10 randomly selected sunflower seeds were recorded. These data are summarised below.

$$\sum w ^ { 2 } = 41252 \quad \sum w p = 27557.8 \quad \sum w = 640 \quad \sum p = 431 \quad \mathrm {~S} _ { p p } = 2.72$$

1. Find the value of \(\mathrm { S } _ { w w }\) and the value of \(\mathrm { S } _ { w p }\)
2. Calculate the product moment correlation coefficient between \(p\) and \(w\)
3. Give an interpretation of your product moment correlation coefficient. The equation of the regression line of \(p\) on \(w\) is given in the form \(p = a + b w\)
4. Find the equation of the regression line of \(p\) on \(w\)
5. Hence estimate the percentage oil content of a sunflower seed which weighs 60 milligrams. \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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