5.09c Calculate regression line

5 A shop manager recorded the maximum daytime temperature \(T ^ { \circ } \mathrm { C }\) and the number \(C\) of ice creams sold on 9 summer days. The results are given in the table and illustrated in the scatter diagram. $$n = 9 , \Sigma t = 232 , \Sigma c = 280 , \Sigma t ^ { 2 } = 6130 , \Sigma c ^ { 2 } = 9444 , \Sigma t c = 7489$$

1. State, with a reason, whether one of the variables \(C\) or \(T\) is likely to be dependent upon the other.
2. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
3. State with a reason what the value of \(r\) would have been if the temperature had been measured in \({ } ^ { \circ } \mathrm { F }\) rather than \({ } ^ { \circ } \mathrm { C }\).
4. Calculate the equation of the least squares regression line of \(c\) on \(t\).
5. The regression line is drawn on the copy of the scatter diagram in the Printed Answer Booklet. Use this diagram to explain what is meant by "least squares".

9 The values of a set of bivariate data \(\left( x _ { i } , y _ { i } \right)\) can be summarised by $$n = 50 , \sum x = 1270 , \sum y = 5173 , \sum x ^ { 2 } = 42767 , \sum y ^ { 2 } = 701301 , \sum x y = 173161 .$$ Ten independent observations of \(Y\) are obtained, all corresponding to \(x = 20\). It may be assumed that the variance of \(Y\) is 1.9 , independently of the value of \(x\). Find a \(95 \%\) confidence interval for the mean \(\bar { Y }\) of the 10 observations of \(Y\). \section*{END OF QUESTION PAPER}

5 The birth rate, \(x\) per thousand members of the population, and the life expectancy at birth, \(y\) years, in 14 randomly selected African countries are given in the table. \(n = 14 , \sum x = 72.8 , \sum y = 826 , \sum x ^ { 2 } = 392.96 , \sum y ^ { 2 } = 48924.54 , \sum x y = 4279.16\)

1. Calculate Pearson's product-moment correlation coefficient \(r\) for the data.
2. State what would be the effect on the value of \(r\) if the birth rate were given per hundred and not per thousand.
3. Explain what the sign of \(r\) tells you about the relationship between life expectancy and birth rate for these countries.
4. Test at the \(5 \%\) significance level whether there is correlation between birth rate and life expectancy at birth in African countries.
5. A researcher wants to estimate the life expectancy at birth in Zimbabwe, where the birth rate is 3.9 per thousand. Explain whether a reliable estimate could be obtained using the regression line of \(y\) on \(x\) for the given data.

1. Tomas is studying the relationship between temperature and hours of sunshine in Seapron. He records the midday temperature, \(t ^ { \circ } \mathrm { C }\), and the hours of sunshine, \(s\) hours, for a random sample of 9 days in October. He calculated the following statistics

$$\sum s = 15 \quad \sum s ^ { 2 } = 44.22 \quad \sum t = 127 \quad \mathrm {~S} _ { t t } = 10.89$$

1. Calculate \(\mathrm { S } _ { s s }\) Tomas calculated the product moment correlation coefficient between \(s\) and \(t\) to be 0.832 correct to 3 decimal places.
2. State, giving a reason, whether or not this correlation coefficient supports the use of a linear regression model to describe the relationship between midday temperature and hours of sunshine.
3. State, giving a reason, why the hours of sunshine would be the explanatory variable in a linear regression model between midday temperature and hours of sunshine.
4. Find \(\mathrm { S } _ { s t }\)
5. Calculate a suitable linear regression equation to model the relationship between midday temperature and hours of sunshine.
6. Calculate the standard deviation of \(s\) Tomas uses this model to estimate the midday temperature in Seapron for a day in October with 5 hours of sunshine.
7. State the value of Tomas' estimate. Given that the values of \(s\) are all within 2 standard deviations of the mean,
8. comment, giving your reason, on the reliability of this estimate.

1. A company wants to pay its employees according to their performance at work. Last year's performance score \(x\) and annual salary \(y\), in thousands of dollars, were recorded for a random sample of 10 employees of the company.

The performance scores were $$\begin{array} { l l l l l l l l l l } 15 & 24 & 32 & 39 & 41 & 18 & 16 & 22 & 34 & 42 \end{array}$$ (You may use \(\sum x ^ { 2 } = 9011\) )

1. Find the mean and the variance of these performance scores. The corresponding \(y\) values for these 10 employees are summarised by $$\sum y = 306.1 \quad \text { and } \quad \mathrm { S } _ { y y } = 546.3$$
2. Find the mean and the variance of these \(y\) values. The regression line of \(y\) on \(x\) based on this sample is $$y = 12.0 + 0.659 x$$
3. Find the product moment correlation coefficient for these data.
4. State, giving a reason, whether or not the value of the product moment correlation coefficient supports the use of a regression line to model the relationship between performance score and annual salary. The company decides to use this regression model to determine future salaries.
5. Find the proposed annual salary, in dollars, for an employee who has a performance score of 35

2. A large company is analysing how much money it spends on paper in its offices each year. The number of employees in the office, \(x\), and the amount spent on paper in a year, \(p\) (\$ hundreds), in each of 12 randomly selected offices were recorded. The results are summarised in the following statistics. $$\sum x = 93 \quad \mathrm {~S} _ { x x } = 148.25 \quad \sum p = 273 \quad \sum p ^ { 2 } = 6602.72 \quad \sum x p = 2347$$

1. Show that \(\mathrm { S } _ { x p } = 231.25\)
2. Find the product moment correlation coefficient for these data.
3. Find the equation of the regression line of \(p\) on \(x\) in the form \(p = a + b x\)
4. Give an interpretation of the gradient of your regression line. The director of the company wants to reduce the amount spent on paper each year. He wants each office to aim for a model of the form \(p = \frac { 4 } { 5 } a + \frac { 1 } { 2 } b x\), where \(a\) and \(b\) are the values found in part (c). Using the data for the 93 employees from the 12 offices,
5. estimate the percentage saving in the amount spent on paper each year by the company using the director's model.

1. Eight students took tests in mathematics and physics. The marks for each student are given in the table below where \(m\) represents the mathematics mark and \(p\) the physics mark.

A science teacher believes that students' marks in physics depend upon their mathematical ability. The teacher decides to investigate this relationship using the test marks.

1. Write down which is the explanatory variable in this investigation.
2. Draw a scatter diagram to illustrate these data.
3. Showing your working, find the equation of the regression line of \(p\) on \(m\).
4. Draw the regression line on your scatter diagram. A ninth student was absent for the physics test, but she sat the mathematics test and scored 15 .
5. Using this model, estimate the mark she would have scored in the physics test.

3 [Figure 1, printed on the insert, is provided for use in this question.]
A parcel delivery company has a depot on the outskirts of a town. Each weekday, a van leaves the depot to deliver parcels across a nearby area. The table below shows, for a random sample of 10 weekdays, the number, \(x\), of parcels to be delivered and the total time, \(y\) minutes, that the van is out of the depot.

1. On Figure 1, plot a scatter diagram of these data.
2. Calculate the equation of the least squares regression line of \(y\) on \(x\) and draw your line on Figure 1.
3. Use your regression equation to estimate the total time that the van is out of the depot when delivering:
   1. 15 parcels;
   2. 35 parcels. Comment on the likely reliability of each of your estimates.
4. The time that the van is out of the depot delivering parcels may be thought of as the time needed to travel to and from the area plus an amount of time proportional to the number of parcels to be delivered. Given that the regression line of \(y\) on \(x\) is of the form \(y = a + b x\), give an interpretation, in context, for each of your values of \(a\) and \(b\).
   (2 marks)

7 [Figure 1, printed on the insert, is provided for use in this question.]
Stan is a retired academic who supplements his pension by mowing lawns for customers who live nearby. As part of a review of his charges for this work, he measures the areas, \(x \mathrm {~m} ^ { 2 }\), of a random sample of eight of his customers' lawns and notes the times, \(y\) minutes, that it takes him to mow these lawns. His results are shown in the table.

1. On Figure 1, plot a scatter diagram of these data.
2. Calculate the equation of the least squares regression line of \(y\) on \(x\). Draw your line on Figure 1.
3. Calculate the value of the residual for Customer H and indicate how your value is confirmed by your scatter diagram.
4. Given that Stan charges \(\pounds 12\) per hour, estimate the charge for mowing a customer's lawn that has an area of \(560 \mathrm {~m} ^ { 2 }\).

3 The table shows, for each of a random sample of 7 weeks, the number of customers, \(x\), who purchased fuel from a filling station, together with the total volume, \(y\) litres, of fuel purchased by these customers.

1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
2. Estimate the volume of fuel sold during a week in which 200 customers purchase fuel.
3. Comment on the likely reliability of your estimate in part (b), given that, for the regression line calculated in part (a), the values of the 7 residuals lie between approximately - 415 litres and + 430 litres.

4 The time taken for a fax machine to scan an A4 sheet of paper is dependent, in part, on the number of lines of print on the sheet. The table below shows, for each of a random sample of 8 sheets of A4 paper, the number, \(x\), of lines of print and the scanning time, \(y\) seconds, taken by the fax machine.

1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
2. The following table lists some of the residuals for the regression line.

   Sheet	\(\mathbf { 1 }\)	\(\mathbf { 2 }\)	\(\mathbf { 3 }\)	\(\mathbf { 4 }\)	\(\mathbf { 5 }\)	\(\mathbf { 6 }\)	\(\mathbf { 7 }\)	\(\mathbf { 8 }\)
   Residual	- 0.174	0.418		0.085	- 0.254	0.906		- 0.157

   1. Calculate the values of the residuals for sheets 3 and 7 .
   2. Hence explain what can be deduced about the regression line.
3. The time, \(z\) seconds, to transmit an A4 page after scanning is given by: $$z = 0.80 + 0.05 x$$ Estimate the total time to scan and transmit an A4 page containing:
   1. 15 lines of print;
   2. 75 lines of print. In each case comment on the likely reliability of your estimate.

3 A new car tyre is fitted to a wheel. The tyre is inflated to its recommended pressure of 265 kPa and the wheel left unused. At 3-month intervals thereafter, the tyre pressure is measured with the following results:

1. 1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
   2. Interpret in context the value for the gradient of your line.
   3. Comment on the value for the intercept with the \(y\)-axis of your line.
2. The tyre manufacturer states that, when one of these new tyres is fitted to the wheel of a car and then inflated to 265 kPa , a suitable regression equation is of the form $$y = 265 + b x$$ The manufacturer also states that, as the car is used, the tyre pressure will decrease at twice the rate of that found in part (a).
   1. Suggest a suitable value for \(b\).
   2. One of these new tyres is fitted to the wheel of a car and inflated to 265 kPa . The car is then used for 8 months, after which the tyre pressure is checked for the first time. Show that, accepting the manufacturer's statements, the tyre pressure can be expected to have fallen below its minimum safety value of 220 kPa .
      (2 marks)

5 The table shows the number of customers, \(x\), and the takings, \(\pounds y\), recorded to the nearest \(\pounds 10\), at a local butcher's shop on each of 10 randomly selected weekdays.

1. The first 6 pairs of data values in this table are plotted on the scatter diagram shown on the opposite page. Plot the final 4 pairs of data values on the scatter diagram.
2. 1. Calculate the equation of the least squares regression line in the form \(y = a + b x\) and draw your line on the scatter diagram.
   2. Interpret your value for \(b\) in the context of the question.
   3. State why your value for \(a\) has no practical interpretation.
3. Estimate, to the nearest \(\pounds 10\), the shop's takings when the number of customers is 50 .
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}]{4c679380-894f-4d36-aec8-296b662058e2-14_1255_1705_1448_155}
   Butcher's shop \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Answer space for question 5} \includegraphics[alt={},max width=\textwidth]{4c679380-894f-4d36-aec8-296b662058e2-15_2335_1760_372_100}
   \end{figure}

4 Stephan is a roofing contractor who is often required to replace loose ridge tiles on house roofs. In order to help him to quote more accurately the prices for such jobs in the future, he records, for each of 11 recently repaired roofs, the number of ridge tiles replaced, \(x _ { i }\), and the time taken, \(y _ { i }\) hours. His results are shown in the table.

1. The pairs of data values for roofs 1 to 7 are plotted on the scatter diagram shown on the opposite page. Plot the 4 pairs of data values for roofs 8 to 11 on the scatter diagram.
2. 1. Calculate the equation of the least squares regression line of \(y _ { i }\) on \(x _ { i }\), and draw your line on the scatter diagram.
   2. Interpret your values for the gradient and for the intercept of this regression line.
3. Estimate the time that it would take Stephan to replace 15 loose ridge tiles on a house roof.
4. Given that \(r _ { i }\) denotes the residual for the point representing roof \(i\) :
   1. calculate the value of \(r _ { 6 }\);
   2. state why the value of \(\sum _ { i = 1 } ^ { 11 } r _ { i }\) gives no useful information about the connection between the number of ridge tiles replaced and the time taken.
      [0pt] [1 mark]
      \section*{Answer space for question 4}
      \includegraphics[max width=\textwidth, alt={}]{6fbb8891-e6de-42fe-a195-ea643552fdcf-11_2385_1714_322_155}

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. \section*{June 2005}

1. A biologist records the length, \(y \mathrm {~cm}\), and the weight, \(w \mathrm {~kg}\), of 50 rabbits. The following summary statistics are calculated from these data.

$$\sum y = 2015 \quad \sum y ^ { 2 } = 81938.5 \quad \sum w = 125 \quad \mathrm {~S} _ { w w } = 72.25 \quad \mathrm {~S} _ { y w } = 219.55$$

1. 1. Show that \(\mathrm { S } _ { y y } = 734\)
   2. Calculate the product moment correlation coefficient for these data. Give your answer to 3 decimal places.
2. Interpret your value of the product moment correlation coefficient. The biologist believes that a linear regression model may be appropriate to describe these data.
3. State, with a reason, whether or not your value of the product moment correlation coefficient is consistent with the biologist’s belief.
4. Find the equation of the regression line of \(w\) on \(y\), giving your answer in the form \(w = a + b y\) Jeff has a pet rabbit of length 45 cm .
5. Use your regression equation to estimate the weight of Jeff's rabbit.

3 At a local athletics club, data on the ages of the members and their times to run a 10 km course are recorded. For a random sample of 25 club members aged between 20 and 60, their ages ( \(x\) years) and times ( \(y\) minutes) are summarised as follows. $$n = 25 \quad \Sigma x = 1002 \quad \Sigma x ^ { 2 } = 43508 \quad \Sigma y = 1865 \quad \Sigma y ^ { 2 } = 142749 \quad \Sigma x y = 77532$$

1. Calculate the product moment correlation coefficient for these data.
2. Show that the equation of the least squares regression line of \(y\) on \(x\) is \(y = 0.83 x + 41.28\), where the coefficients are given correct to 2 decimal places.
3. Use the equation given in part (ii) to estimate the time taken by someone who is
   1. 50 years old,
   2. 65 years old. Comment on the validity of each of these estimates.

2 The table shows the turnover, in millions of pounds, of a small company at 3-year intervals over a period of 15 years, starting in 2000.

1. For the information in the table find the equation of the least squares regression line of \(y\) on \(x\), where \(x\) is the year since 2000 and \(y\) is the turnover in millions of pounds.
2. Use the equation of the regression line to calculate the residual for 2009.
3. Use the equation of the regression line to estimate the turnover in 2024, and explain why it is inadvisable to rely on this estimate.

2 A teacher is monitoring the progress of students. The length of time, \(x\) hours, spent revising in a given week is compared to the score, \(y\), achieved in an assessment at the end of the week. The scatter diagram for a random sample of 8 students is shown below. \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-2_866_967_715_589} The data are summarised as \(\Sigma x = 24.6 , \Sigma y = 404 , \Sigma x ^ { 2 } = 105.56 , \Sigma y ^ { 2 } = 20820\) and \(\Sigma x y = 1350.2\).

1. Find the equation of the least squares regression line of \(y\) on \(x\).
2. Calculate the product moment correlation coefficient for the data.
3. A ninth student, Jane, revises for 1.5 hours.
   1. Estimate her score in the assessment.
   2. Comment on the reliability of this estimate.

13 A seed company investigated how well African Marigold seeds germinated when the seeds were past their sell-by date. The table shows the average number of seeds which germinated per packet, \(y\), and the number of months past their sell-by date, \(t\). The summary data for the investigation were as follows. $$\Sigma t = 150 \quad \Sigma t ^ { 2 } = 5500 \quad \Sigma y = 101.2 \quad \Sigma y ^ { 2 } = 2146.86 \quad \Sigma t y = 2740$$

1. Calculate the equation of the regression line of \(y\) on \(t\).
2. Use your regression line to calculate \(y\) when \(t = 10\). Compare your answer with the value of \(y\) when \(t = 10\) in the table and comment on the result.
3. Use your regression line to calculate \(y\) when \(t = 100\). Comment on the validity of this result.
4. Suggest with reasons whether the regression line provides a good model for predicting the germination of seeds past their sell-by date.

A set of \(20\) pairs of bivariate data \((x, y)\) is summarised by $$\Sigma x = 200, \quad \Sigma x^2 = 2125, \quad \Sigma y = 240, \quad \Sigma y^2 = 8245.$$ The product moment correlation coefficient is \(-0.992\).

1. What does the value of the product moment correlation coefficient indicate about a scatter diagram of the data points? [1]
2. Find the equation of the regression line of \(y\) on \(x\). [6]
3. The equation of the regression line of \(x\) on \(y\) is \(x = a' + b'y\). Find the value of \(b'\). [2]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{3l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

Answer only one of the following two alternatives. EITHER A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(4mg\) and natural length \(l\). The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\). The particle is pulled down a vertical distance \(\frac{3l}{4}\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic with period \(\pi\sqrt{\left(\frac{l}{g}\right)}\). [4] At an instant when \(P\) is moving vertically downwards through \(E\), the string is cut. When \(P\) has descended a further distance \(\frac{5l}{4}\) under gravity, it strikes a fixed smooth plane which is inclined at 30° to the horizontal. The coefficient of restitution between \(P\) and the plane is \(\frac{1}{3}\). Show that the speed of \(P\) immediately after the impact is \(\frac{1}{3}\sqrt{(5gl)}\). [8] OR A new restaurant \(S\) has recently opened in a particular town. In order to investigate any effect of \(S\) on an existing restaurant \(R\), the daily takings, \(x\) and \(y\) in thousands of dollars, at \(R\) and \(S\) respectively are recorded for a random sample of 8 days during a six-month period. The results are shown in the following table.

Day	1	2	3	4	5	6	7	8
\(x\)	1.2	1.4	0.9	1.1	0.8	1.0	0.6	1.5
\(y\)	0.3	0.4	0.6	0.6	0.25	0.75	0.6	0.35

1. Calculate the product moment correlation coefficient for this sample. [4]
2. Stating your hypotheses, test, at the 2.5\% significance level, whether there is negative correlation between daily takings at the two restaurants and comment on your result in the context of the question. [5]

Another sample is taken over \(N\) randomly chosen days and the product moment correlation coefficient is found to be \(-0.431\). A test, at the 5\% significance level, shows that there is evidence of negative correlation between daily takings in the two restaurants.

1. Find the range of possible values of \(N\). [3]

A random sample of 5 pairs of values \((x, y)\) is given in the following table.

1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\). [4]
2. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample. [3]
3. Test, at the 10% significance level, whether there is evidence of non-zero correlation between the variables. [4]

A random sample of 5 pairs of values \((x, y)\) is given in the following table.

\(x\)	1	2	4	5	8
\(y\)	7	5	8	6	4

1. Find, showing all necessary working, the equation of the regression line of \(y\) on \(x\). [4]
2. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample. [3]
3. Test, at the 10% significance level, whether there is evidence of non-zero correlation between the variables. [4]