5.09b Least squares regression: concepts

7. An ice cream seller believes that there is a relationship between the temperature on a summer day and the number of ice creams sold. Over a period of 10 days he records the temperature at 1 p.m., \(t ^ { \circ } \mathrm { C }\), and the number of ice creams sold, \(c\), in the next hour. The data he collects is summarised in the table below. [Use \(\left. \Sigma t ^ { 2 } = 3025 , \Sigma c ^ { 2 } = 14245 , \Sigma c t = 6526 .\right]\)

1. Calculate the value of the product moment correlation coefficient between \(t\) and \(c\).
2. State whether or not your value supports the use of a regression equation to predict the number of ice creams sold. Give a reason for your answer.
3. Find the equation of the least squares regression line of \(c\) on \(t\) in the form \(c = a + b t\).
4. Interpret the value of \(b\).
5. Estimate the number of ice creams sold between 1 p.m. and 2 p.m. when the temperature at 1 p.m. is \(16 ^ { \circ } \mathrm { C }\).
   (3)
6. At 1 p.m. on a particular day, the highest temperature for 50 years was recorded. Give a reason why you should not use the regression equation to predict ice cream sales on that day.
   (1)

2. A researcher thinks there is a link between a person's height and level of confidence. She measured the height \(h\), to the nearest cm , of a random sample of 9 people. She also devised a test to measure the level of confidence \(c\) of each person. The data are shown in the table below. [You may use \(\Sigma h ^ { 2 } = 272094 , \Sigma c ^ { 2 } = 2878966 , \Sigma h c = 884484\) ]

1. Draw a scatter diagram to illustrate these data.
2. Find exact values of \(S _ { h c } S _ { h h }\) and \(S _ { c c }\).
3. Calculate the value of the product moment correlation coefficient for these data.
4. Give an interpretation of your correlation coefficient.
5. Calculate the equation of the regression line of \(c\) on \(h\) in the form \(c = a + b h\).
6. Estimate the level of confidence of a person of height 180 cm .
7. State the range of values of \(h\) for which estimates of \(c\) are reliable.

1. A long distance lorry driver recorded the distance travelled, \(m\) miles, and the amount of fuel used, \(f\) litres, each day. Summarised below are data from the driver's records for a random sample of 8 days.

The data are coded such that \(x = m - 250\) and \(y = f - 100\). $$\Sigma x = 130 \quad \Sigma y = 48 \quad \Sigma x y = 8880 \quad \mathrm {~S} _ { x x } = 20487.5$$

1. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\).
2. Hence find the equation of the regression line of \(f\) on \(m\).
3. Predict the amount of fuel used on a journey of 235 miles.

1. A metallurgist measured the length, \(l \mathrm {~mm}\), of a copper rod at various temperatures, \(t ^ { \circ } \mathrm { C }\), and recorded the following results.

The results were then coded such that \(x = t\) and \(y = l - 2460.00\).

1. Calculate \(S _ { x y }\) and \(S _ { x x }\).
   (You may use \(\Sigma x ^ { 2 } = 15965.01\) and \(\Sigma x y = 757.467\) )
2. Find the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\).
3. Estimate the length of the rod at \(40 ^ { \circ } \mathrm { C }\).
4. Find the equation of the regression line of \(l\) on \(t\).
5. Estimate the length of the rod at \(90 ^ { \circ } \mathrm { C }\).
6. Comment on the reliability of your estimate in part (e).

3. A student is investigating the relationship between the price ( \(y\) pence) of 100 g of chocolate and the percentage ( \(x \%\) ) of cocoa solids in the chocolate.
The following data is obtained (You may use: \(\sum x = 315 , \sum x ^ { 2 } = 15225 , \sum y = 620 , \sum y ^ { 2 } = 56550 , \sum x y = 28750\) )

1. On the graph paper on page 9 draw a scatter diagram to represent these data.
2. Show that \(S _ { x y } = 4337.5\) and find \(S _ { x x }\). The student believes that a linear relationship of the form \(y = a + b x\) could be used to describe these data.
3. Use linear regression to find the value of \(a\) and the value of \(b\), giving your answers to 1 decimal place.
4. Draw the regression line on your scatter diagram. The student believes that one brand of chocolate is overpriced.
5. Use the scatter diagram to
   1. state which brand is overpriced,
   2. suggest a fair price for this brand. Give reasons for both your answers.
      \includegraphics[max width=\textwidth, alt={}]{045e10d2-1766-4399-aa0a-5619dd0cce0f-06_2454_1485_282_228}
      The data on page 8 has been repeated here to help you

      Chocolate brand	A	\(B\)	\(C\)	D	\(E\)	\(F\)	G	\(H\)
      \(x\) (\% cocoa)	10	20	30	35	40	50	60	70
      \(y\) (pence)	35	55	40	100	60	90	110	130

      (You may use: \(\sum x = 315 , \sum x ^ { 2 } = 15225 , \sum y = 620 , \sum y ^ { 2 } = 56550 , \sum x y = 28750\) )

4. Crickets make a noise. The pitch, \(v \mathrm { kHz }\), of the noise made by a cricket was recorded at 15 different temperatures, \(t ^ { \circ } \mathrm { C }\). These data are summarised below. $$\sum t ^ { 2 } = 10922.81 , \sum v ^ { 2 } = 42.3356 , \sum t v = 677.971 , \sum t = 401.3 , \sum v = 25.08$$

1. Find \(S _ { t t } , S _ { v v }\) and \(S _ { t v }\) for these data.
2. Find the product moment correlation coefficient between \(t\) and \(v\).
3. State, with a reason, which variable is the explanatory variable.
4. Give a reason to support fitting a regression model of the form \(v = a + b t\) to these data.
5. Find the value of \(a\) and the value of \(b\). Give your answers to 3 significant figures.
6. Using this model, predict the pitch of the noise at \(19 ^ { \circ } \mathrm { C }\).

5. The weight, \(w\) grams, and the length, \(l \mathrm {~mm}\), of 10 randomly selected newborn turtles are given in the table below. $$\text { (You may use } \mathrm { S } _ { l l } = 33.381 \quad \mathrm {~S} _ { w l } = 59.99 \quad \mathrm {~S} _ { w w } = 120.1 \text { ) }$$

1. Find the equation of the regression line of \(w\) on \(l\) in the form \(w = a + b l\).
2. Use your regression line to estimate the weight of a newborn turtle of length 60 mm .
3. Comment on the reliability of your estimate giving a reason for your answer.

6. A travel agent sells flights to different destinations from Beerow airport. The distance \(d\), measured in 100 km , of the destination from the airport and the fare \(\pounds f\) are recorded for a random sample of 6 destinations. $$\text { [You may use } \sum d ^ { 2 } = 152.09 \quad \sum f ^ { 2 } = 3686 \quad \sum f d = 723.1 \text { ] }$$

1. Using the axes below, complete a scatter diagram to illustrate this information.
2. Explain why a linear regression model may be appropriate to describe the relationship between \(f\) and \(d\).
3. Calculate \(S _ { d d }\) and \(S _ { f d }\)
4. Calculate the equation of the regression line of \(f\) on \(d\) giving your answer in the form \(f = a + b d\).
5. Give an interpretation of the value of \(b\). Jane is planning her holiday and wishes to fly from Beerow airport to a destination \(t \mathrm {~km}\) away. A rival travel agent charges 5 p per km.
6. Find the range of values of \(t\) for which the first travel agent is cheaper than the rival. \includegraphics[max width=\textwidth, alt={}, center]{039e6fcf-3222-40cc-95ea-37b8dc4a4ddb-11_1013_1701_1718_116}

3. A scientist is researching whether or not birds of prey exposed to pollutants lay eggs with thinner shells. He collects a random sample of egg shells from each of 6 different nests and tests for pollutant level, \(p\), and measures the thinning of the shell, \(t\). The results are shown in the table below. [You may use \(\sum p ^ { 2 } = 1967\) and \(\sum p t = 694\) ]

1. Draw a scatter diagram on the axes on page 7 to represent these data.
2. Explain why a linear regression model may be appropriate to describe the relationship between \(p\) and \(t\).
3. Calculate the value of \(S _ { p t }\) and the value of \(S _ { p p }\).
4. Find the equation of the regression line of \(t\) on \(p\), giving your answer in the form \(t = a + b p\).
5. Plot the point ( \(\bar { p } , \bar { t }\) ) and draw the regression line on your scatter diagram. The scientist reviews similar studies and finds that pollutant levels above 16 are likely to result in the death of a chick soon after hatching.
6. Estimate the minimum thinning of the shell that is likely to result in the death of a chick. \includegraphics[max width=\textwidth, alt={}, center]{0593544d-392d-465b-b922-c9cb1435abb5-05_1257_1568_301_173}

1. A meteorologist believes that there is a relationship between the height above sea level, \(h \mathrm {~m}\), and the air temperature, \(t ^ { \circ } \mathrm { C }\). Data is collected at the same time from 9 different places on the same mountain. The data is summarised in the table below.

[You may assume that \(\sum h = 7150 , \sum t = 110 , \sum h ^ { 2 } = 7171500 , \sum t ^ { 2 } = 1716\), \(\sum t h = 64980\) and \(\mathrm { S } _ { t t } = 371.56\) ]

1. Calculate \(\mathrm { S } _ { t h }\) and \(\mathrm { S } _ { h h }\). Give your answers to 3 significant figures.
2. Calculate the product moment correlation coefficient for this data.
3. State whether or not your value supports the use of a regression equation to predict the air temperature at different heights on this mountain. Give a reason for your answer.
4. Find the equation of the regression line of \(t\) on \(h\) giving your answer in the form \(t = a + b h\).
5. Interpret the value of \(b\).
6. Estimate the difference in air temperature between a height of 500 m and a height of 1000 m .

3. A large company is analysing how much money it spends on paper in its offices every year. The number of employees, \(x\), and the amount of money spent on paper, \(p\) ( \(\pounds\) hundreds), in 8 randomly selected offices are given in the table below. $$\text { (You may use } \sum x ^ { 2 } = 1160 \quad \sum p = 299.2 \quad \sum p ^ { 2 } = 11422 \quad \sum x p = 3449.5 \text { ) }$$

1. Show that \(S _ { p p } = 231.92\) and find the value of \(S _ { x x }\) and the value of \(S _ { x p }\)
2. Calculate the product moment correlation coefficient between \(x\) and \(p\). The equation of the regression line of \(p\) on \(x\) is given in the form \(p = a + b x\).
3. Show that, to 3 significant figures, \(b = 0.824\) and find the value of \(a\).
4. Estimate the amount of money spent on paper in an office with 10 employees.
5. Explain the effect each additional employee has on the amount of money spent on paper. Later the company realised it had made a mistake in adding up its costs, \(p\). The true costs were actually half of the values recorded. The product moment correlation coefficient and the equation of the linear regression line are recalculated using this information.
6. Write down the new value of
   1. the product moment correlation coefficient,
   2. the gradient of the regression line.

3. The table shows data on the number of visitors to the UK in a month, \(v\) (1000s), and the amount of money they spent, \(m\) ( \(\pounds\) millions), for each of 8 months. You may use \(S _ { v v } = 42587.5 \quad S _ { v m } = 31512.5 \quad S _ { m m } = 25187.5 \quad \sum v = 19390 \quad \sum m = 10610\)

1. Find the product moment correlation coefficient between \(m\) and \(v\).
2. Give a reason to support fitting a regression model of the form \(m = a + b v\) to these data.
3. Find the value of \(b\) correct to 3 decimal places.
4. Find the equation of the regression line of \(m\) on \(v\).
5. Interpret your value of \(b\).
6. Use your answer to part (d) to estimate the amount of money spent when the number of visitors to the UK in a month is 2500000
7. Comment on the reliability of your estimate in part (f). Give a reason for your answer.

6. To test the heating of tyre material, tyres are run on a test rig at chosen speeds under given conditions of load, pressure and surrounding temperature. The following table gives values of \(x\), the test rig speed in miles per hour (mph), and the temperature, \(y ^ { \circ } \mathrm { C }\), generated in the shoulder of the tyre for a particular tyre material.

1. Draw a scatter diagram to represent these data.
2. Give a reason to support the fitting of a regression line of the form \(y = a + b x\) through these points.
3. Find the values of \(a\) and \(b\).
   (You may use \(\Sigma x ^ { 2 } = 9500 , \Sigma y ^ { 2 } = 45483 , \Sigma x y = 20615\) )
4. Give an interpretation for each of \(a\) and \(b\).
5. Use your line to estimate the temperature at 50 mph and explain why this estimate differs from the value given in the table. A tyre specialist wants to estimate the temperature of this tyre material at 12 mph and 85 mph .
6. Explain briefly whether or not you would recommend the specialist to use this regression equation to obtain these estimates.

1. A company wants to pay its employees according to their performance at work. The performance score \(x\) and the annual salary, \(y\) in \(\pounds 100\) s, for a random sample of 10 of its employees for last year were recorded. The results are shown in the table below.

$$\text { [You may assume } \left. \Sigma x y = 69798 , \Sigma x ^ { 2 } = 7266 \right]$$

1. Draw a scatter diagram to represent these data.
2. Calculate exact values of \(S _ { x y }\) and \(S _ { x x }\).
3. 1. Calculate the equation of the regression line of \(y\) on \(x\), in the form \(y = a + b x\). Give the values of \(a\) and \(b\) to 3 significant figures.
   2. Draw this line on your scatter diagram.
4. Interpret the gradient of the regression line. The company decides to use this regression model to determine future salaries.
5. Find the proposed annual salary for an employee who has a performance score of 35 .

2. An experiment carried out by a student yielded pairs of \(( x , y )\) observations such that $$\bar { x } = 36 , \quad \bar { y } = 28.6 , \quad S _ { x x } = 4402 , \quad S _ { x y } = 3477.6$$

1. Calculate the equation of the regression line of \(y\) on \(x\) in the form \(y = a + b x\). Give your values of \(a\) and \(b\) to 2 decimal places.
2. Find the value of \(y\) when \(x = 45\).

1 At a certain small restaurant, the waiting time is defined as the time between sitting down at a table and a waiter first arriving at the table. This waiting time is dependent upon the number of other customers already seated in the restaurant. Alex is a customer who visited the restaurant on 10 separate days. The table shows, for each of these days, the number, \(x\), of customers already seated and his waiting time, \(y\) minutes.

1. Calculate the equation of the least squares regression line of \(y\) on \(x\) in the form \(y = a + b x\).
2. Give an interpretation, in context, for each of your values of \(a\) and \(b\).
3. Use your regression equation to estimate Alex's waiting time when the number of customers already seated in the restaurant is:
   1. 5 ;
   2. 25 .
4. Comment on the likely reliability of each of your estimates in part (c), given that, for the regression line calculated in part (a), the values of the 10 residuals lie between + 1.1 minutes and - 1.1 minutes.

4 [Figure 1, printed on the insert, is provided for use in this question.]
Roseen is a self-employed decorator who wishes to estimate the times that it will take her to decorate bedrooms based upon their floor areas. She records the floor area, \(x \mathrm {~m} ^ { 2 }\), and the decorating time, \(y\) hours, for each of 10 bedrooms she has recently decorated.

1. On Figure 1, plot a scatter diagram of these data.
2. Calculate the equation of the least squares regression line of \(y\) on \(x\).
3. Draw your regression line on Figure 1.
4. 1. Use your regression equation to estimate the time that Roseen will take to decorate a bedroom with a floor area of \(15 \mathrm {~m} ^ { 2 }\).
   2. Making reference to Figure 1, comment on the likely reliability of your estimate in part (d)(i).

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.

5 Craig uses his car to travel regularly from his home to the area hospital for treatment. He leaves home at \(x\) minutes after 7.30 am and then takes \(y\) minutes to arrive at the hospital's reception desk. His results for 11 mornings are shown in the table.

1. Explain why the time taken by Craig between leaving home and arriving at the hospital's reception desk is the response variable.
2. Calculate the equation of the least squares regression line of \(y\) on \(x\), writing your answer in the form \(y = a + b x\).
3. On a particular day, Craig needs to arrive at the hospital's reception desk no later than 9.00 am . He leaves home at 7.45 am . Estimate the number of minutes before 9.00 am that Craig will arrive at the hospital's reception desk. Give your answer to the nearest minute.
4. 1. Use your equation to estimate \(y\) when \(x = 85\).
   2. Give one statistical reason and one reason based on the context of this question as to why your estimate in part (d)(i) is unlikely to be realistic.埗

5 An experiment was undertaken to collect information on the burning of a specific type of wood as a source of energy. At given fixed levels of the wood's moisture content, \(x\) per cent, its corresponding calorific value, \(y \mathrm { MWh } /\) tonne, on burning was determined. The results are shown in the table.

1. Explain why calorific value is the response variable.
2. Calculate the equation of the least squares regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\).
3. Interpret, in context, your values for \(a\) and \(b\).
4. Use your equation to estimate the wood's calorific value when it has a moisture content of 27 per cent.
5. Calculate the value of the residual for the point \(( 35,2.5 )\).
6. Given that the values of the 13 residuals lie between - 0.28 and + 0.23 , comment on the likely accuracy of your estimate in part (d).
7. 1. Give a general reason why your equation should not be used to estimate the wood's calorific value when it has a moisture content of 80 per cent.
   2. Give a specific reason, based on the context of this question and with numerical support, why your equation cannot be used to estimate the wood's calorific value when it has a moisture content of 80 per cent.

1 Bob, a church warden, decides to investigate the lifetime of a particular manufacturer's brand of beeswax candle. Each candle is 30 cm in length. From a box containing a large number of such candles, he selects one candle at random. He lights the candle and, after it has burned continuously for \(x\) hours, he records its length, \(y \mathrm {~cm}\), to the nearest centimetre. His results are shown in the table.

1. State the value that you would expect for \(a\) in the equation of the least squares regression line, \(y = a + b x\).
2. 1. Calculate the equation of the least squares regression line, \(y = a + b x\).
   2. Interpret the value that you obtain for \(b\).
   3. It is claimed by the candle manufacturer that the total length of time that such candles are likely to burn for is more than 50 hours. Comment on this claim, giving a numerical justification for your answer.

5 Bob, a gardener, measures the time taken, \(y\) minutes, for 60 grams of weedkiller pellets to dissolve in 10 litres of water at different set temperatures, \(x ^ { \circ } \mathrm { C }\). His results are shown in the table.

1. State why the explanatory variable is temperature.
2. Calculate the equation of the least squares regression line \(y = a + b x\).
3. 1. Interpret, in the context of this question, your value for \(b\).
   2. Explain why no sensible practical interpretation can be given for your value of \(a\).
4. 1. Estimate the time taken to dissolve 60 grams of weedkiller pellets in 10 litres of water at \(30 ^ { \circ } \mathrm { C }\).
   2. Show why the equation cannot be used to make a valid estimate of the time taken to dissolve 60 grams of weedkiller pellets in 10 litres of water at \(75 ^ { \circ } \mathrm { C }\). (2 marks)

1 The table shows the times taken, \(y\) minutes, for a wood glue to dry at different air temperatures, \(x ^ { \circ } \mathrm { C }\).

1. Calculate the equation of the least squares regression line \(y = a + b x\).
2. Estimate the time taken for the glue to dry when the air temperature is \(21 ^ { \circ } \mathrm { C }\).

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}