5.09a Dependent/independent variables

7 An experiment was carried out to determine how much weedkiller to apply per \(100 \mathrm {~m} ^ { 2 }\) in a large field. Ten \(100 \mathrm {~m} ^ { 2 }\) areas of the field were randomly chosen and sprayed with predetermined volumes of the weedkiller. The volume of the weedkiller is denoted by \(x\) litres and the number of weeds that survived is denoted by \(y\). The results are given in the table. $$\left[ \Sigma x = 3.25 , \Sigma x ^ { 2 } = 1.2625 , \Sigma y = 268 , \Sigma y ^ { 2 } = 9548 , \Sigma x y = 66.10 . \right]$$ It is given that the product moment correlation coefficient for the data is - 0.951 , correct to 3 decimal places.

1. Calculate the equation of a suitable regression line, giving a reason for your choice of line.
2. Estimate the best volume of weedkiller to apply, and comment on the reliability of your estimate.

6 Six pairs of values of variables \(x\) and \(y\) are measured. Draw a sketch of a possible scatter diagram of the data for each of the following cases:

1. the product moment correlation coefficient is approximately zero;
2. the product moment correlation coefficient is exactly - 1 . On your diagram for part (i), sketch the regression line of \(y\) on \(x\) and the regression line of \(x\) on \(y\), labelling each line. On your diagram for part (ii), sketch the regression line of \(y\) on \(x\) and state its relationship to the regression line of \(x\) on \(y\).

7 For a random sample of 10 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = 3.25 x - 4.27\). The sum of the ten \(x\) values is 15.6 and the product moment correlation coefficient for the sample is 0.56 . Find the equation of the regression line of \(x\) on \(y\). Test, at the \(5 \%\) significance level, whether there is evidence of non-zero correlation between the variables.

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.

8 The equations of the regression lines for a random sample of 25 pairs of data \(( x , y )\) from a bivariate population are $$\begin{array} { c c } y \text { on } x : & y = 1.28 - 0.425 x , \\ x \text { on } y : & x = 1.05 - 0.516 y . \end{array}$$

1. Find the sample means, \(\bar { x }\) and \(\bar { y }\).
2. Find the product moment correlation coefficient for the sample.
3. Test at the \(5 \%\) significance level whether the population correlation coefficient differs from zero.

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.

9 For a random sample of 10 observations of pairs of values \(( x , y )\), the equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\) are $$y = 4.21 x - 0.862 \quad \text { and } \quad x = 0.043 y + 6.36$$ respectively.

1. Find the value of the product moment correlation coefficient for the sample.
2. Test, at the \(10 \%\) significance level, whether there is evidence of non-zero correlation between the variables.
3. Find the mean values of \(x\) and \(y\) for this sample.
4. Estimate the value of \(x\) when \(y = 2.3\) and comment on the reliability of your answer.

9 A random sample of 10 pairs of values of \(x\) and \(y\) is given in the following table.

1. Find the equation of the regression line of \(y\) on \(x\).
2. Find the product moment correlation coefficient for the sample.
3. Find the estimated value of \(y\) when \(x = 10\), and comment on the reliability of this estimate.
4. Another sample of \(N\) pairs of data from the same population has the same product moment correlation coefficient as the first sample given. A test, at the \(1 \%\) significance level, on this second sample indicates that there is sufficient evidence to conclude that there is positive correlation. Find the set of possible values of \(N\).

9 A random sample of 8 students is chosen from those sitting examinations in both Mathematics and French. Their marks in Mathematics, \(x\), and in French, \(y\), are summarised as follows. $$\Sigma x = 472 \quad \Sigma x ^ { 2 } = 29950 \quad \Sigma y = 400 \quad \Sigma y ^ { 2 } = 21226 \quad \Sigma x y = 24879$$ Another student scored 72 marks in the Mathematics examination but was unable to sit the French examination.

1. Estimate the mark that this student would have obtained in the French examination.
2. Test, at the \(5 \%\) significance level, whether there is non-zero correlation between marks in Mathematics and marks in French.

1. A sixth form college has 84 students in Year 12 and 56 students in Year 13

The head teacher selects a stratified sample of 40 students, stratified by year group.

1. Describe how this sample could be taken. The head teacher is investigating the relationship between the amount of sleep, s hours, that each student had the night before they took an aptitude test and their performance in the test, \(p\) marks.
   For the sample of 40 students, he finds the equation of the regression line of \(p\) on \(s\) to be $$p = 26.1 + 5.60 s$$
2. With reference to this equation, describe the effect that an extra 0.5 hours of sleep may have, on average, on a student's performance in the aptitude test.
3. Describe one limitation of this regression model.

1. The relationship between two variables \(p\) and \(t\) is modelled by the regression line with equation

$$p = 22 - 1.1 t$$ The model is based on observations of the independent variable, \(t\), between 1 and 10

1. Describe the correlation between \(p\) and \(t\) implied by this model. Given that \(p\) is measured in centimetres and \(t\) is measured in days,
2. state the units of the gradient of the regression line. Using the model,
3. calculate the change in \(p\) over a 3-day period. Tisam uses this model to estimate the value of \(p\) when \(t = 19\)
4. Comment, giving a reason, on the reliability of this estimate.

10 Fig. 10.1 shows a sample collected from the large data set. BMI is defined as \(\frac { \text { mass of person in kilograms } } { \text { square of person's height in metres } }\). \begin{table}[h] \end{table}

1. Calculate the mass in kg of a person with a BMI of 23.56 and a height of 181.6 cm , giving your answer correct to 1 decimal place. Fig. 10.2 shows a scatter diagram of BMI against age for the data in the table. A line of best fit has also been drawn. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c08a2212-3104-425e-8aee-7f2d46f23924-09_682_1212_351_248} \captionsetup{labelformat=empty} \caption{Fig. 10.2}
   \end{figure}
2. Describe the correlation between age and BMI.
3. Use the line of best fit to estimate the BMI of a 30-year-old man.
4. Explain why it would not be sensible to use the line of best fit to estimate the BMI of a 60-year-old man.
5. Use your knowledge of the large data set to suggest two reasons why the sample data in the table may not be representative of the population.
6. Once the data in the large data set had been cleaned there were 196 values available for selection. Describe how a sample of size 12 could be generated using systematic sampling so that each of the 196 values could be selected in the sample.

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.

11 The pre-release material contains information concerning median house prices over the period 2004-2015. A spreadsheet has been used to generate a time series graph for two areas: the London borough of "Barking and Dagenham" and "North West". This is shown together with the raw data in Fig. 11.1. \begin{figure}[h] \end{figure} Dr Procter suggests that it is unusual for median house prices in a London borough to be consistently higher than those in other parts of the country.

1. Use your knowledge of the large data set to comment on Dr Procter's suggestion. Dr Procter wishes to predict the median house price in Barking and Dagenham in 2016. She uses the spreadsheet function LINEST to find the equation of the line of best fit for the given data. She obtains the equation \(P = 4897 Y - 9657847\), where \(P\) is the median house price in pounds and \(Y\) is the calendar year, for example 2015.
2. Use Dr Procter's equation to predict the median house price in Barking and Dagenham in
   Professor Jackson uses a simpler model by using the data from 2014 and 2015 only to form a straight-line model.
3. Find the equation Professor Jackson uses in her model.
4. Use Professor Jackson's equation to predict the median house price in Barking and Dagenham in
   Professor Jackson carries out some research online. She finds some information about median house prices in Barking and Dagenham, which is shown in Fig. 11.2. \begin{table}[h]

   2016	2017
   \(\pounds 290000\)	\(\pounds 300000\)

   \captionsetup{labelformat=empty} \caption{Fig. 11.2}
   \end{table}
5. Comment on how well

7 An environmentalist measures the mean concentration, \(c\) milligrams per litre, of a particular chemical in a group of rivers, and the mean mass, \(m\) pounds, of fish of a certain species found in those rivers. The results are given in the table.

1. State which, if either, of \(m\) and \(c\) is an independent variable.
2. Calculate the equation of the least squares regression line of \(c\) on \(m\).
3. State what effect, if any, there would be on your answer to part (ii) if the masses of the fish had been recorded in kilograms rather than pounds. ( \(1 \mathrm {~kg} \approx 2.2\) pounds.)
4. The data is illustrated in the scatter diagram. Explain what is meant by 'least squares', illustrating your answer using the copy of this diagram in the Printed Answer Booklet.
   
   [diagram]

1 A geography student chose a certain point in a stream and took measurements of the speed of flow, \(v \mathrm {~ms} ^ { - 1 }\), of water at various depths, \(d \mathrm {~m}\), below the surface at that point. The results are shown in the table. \(n = 9 \quad \sum d = 2.7 \quad \sum v = 9.0 \quad \sum d ^ { 2 } = 0.96 \quad \sum v ^ { 2 } = 10.4 \quad \sum \mathrm {~d} v = 2.85\)

1. 1. Explain why \(d\) is an example of an independent, controlled variable.
   2. Use two relevant terms to describe the variable \(v\) in a similar way. A statistician believes that the point ( \(0.5,0.4\) ) may be an anomaly.
2. Calculate the equation of the least squares regression line of \(v\) on \(d\) for all the points in the table apart from ( \(0.5,0.4\) ).
3. Use the equation of the line found in part (b) to estimate the value of \(v\) when \(d = 0.5\).
4. Use your answer to part (c) to comment on the statistician's belief.
5. Use the diagram in the Printed Answer Booklet (which does not illustrate the data in this question) to explain what is meant by "least squares regression line".

3 An insurance company collected data concerning the age, \(x\) years, of policy holders and the average size of claim, \(\pounds y\) thousand. The data is summarised as follows. \(n = 32 \quad \sum x = 1340 \quad \sum y = 612 \quad \sum x ^ { 2 } = 64282 \quad \sum y ^ { 2 } = 13418 \quad \sum x y = 27794\)

1. Find the variance of \(x\).
2. Find the equation of the regression line of \(y\) on \(x\).
3. Hence estimate the expected size of claim from a policy holder of age 48. Tom is aged 48. He claims that the range of the data probably does not include people of his age because the mean age for the data is 41.875 , and 48 is not close to this.
4. Use your answer to part (a) to determine how likely it is that Tom's claim is correct.
5. Comment on the reliability of your estimate in part (c). You should refer to the value of the product-moment correlation coefficient for the data, which is 0.579 correct to 3 significant figures.

3 The ages, \(x\) years, and the reaction time, \(t\) seconds, in an experiment carried out on a sample of 15 volunteers are summarised as follows. \(n = 15 \quad \sum x = 762 \quad \sum t = 8.7 \quad \sum x ^ { 2 } = 44204 \quad \sum t ^ { 2 } = 5.65 \quad \sum x t = 490.1\)

1. Calculate the value of the product moment correlation coefficient between \(x\) and \(t\).
2. Calculate the equation of the line of regression of \(t\) on \(x\). Give your answer in the form \(\mathrm { t } = \mathrm { a } + \mathrm { bx }\) where \(a\) and \(b\) are constants to be determined.
3. Explain the relevance of the quantity \(\sum ( t - a - b x ) ^ { 2 }\) to your answer to part (b).
4. Estimate the reaction time, in seconds, for a volunteer aged 42. It is subsequently decided to measure the reaction time in tenths of a second rather than in seconds (so, for example, a time of 0.6 seconds would now be recorded as 6 ).
5. 1. State what effect, if any, this change would have on your answer to part (a).
   2. State what effect, if any, this change would have on your answer to part (b). It is known that the sample of 15 volunteers consisted almost entirely of students and retired people.
6. Using this information, and the value of the product moment correlation coefficient, comment on the reliability of your estimate in part (d).

3 An investor obtains data about the profits of 8 randomly chosen investment accounts over two one-year periods. The profit in the first year for each account is \(p \%\) and the profit in the second year for each account is \(q \%\). The results are shown in the table and in the scatter diagram. \(n = 8 \quad \sum \mathrm { p } = 28.5 \quad \sum \mathrm { q } = 26.7 \quad \sum \mathrm { p } ^ { 2 } = 136.35 \quad \sum \mathrm { q } ^ { 2 } = 116.35 \quad \sum \mathrm { pq } = 116.70\) \includegraphics[max width=\textwidth, alt={}, center]{bf1468d1-e02e-47d2-bf41-5bc8f5b4d7c4-3_782_1280_998_242}

1. State which, if either, of the variables \(p\) and \(q\) is independent.
2. Calculate the equation of the regression line of \(q\) on \(p\).
3. 1. Use the regression line to estimate the value of \(q\) for an investment account for which \(p = 2.5\).
   2. Give two reasons why this estimate could be considered reliable.
4. Comment on the reliability of using the regression line to predict the value of \(q\) when \(p = 7.0\).

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 The table below shows the typical stopping distances \(d\) metres for a particular car travelling at \(v\) miles per hour.

1. State each of the following words that describe the variable \(v\). \section*{Independent Dependent Controlled Response}
2. Calculate the equation of the regression line of \(d\) on \(v\).
3. Use the equation found in part (ii) to estimate the typical stopping distance when this car is travelling at 45 miles per hour. It is given that the product moment correlation coefficient for the data is 0.990 correct to three significant figures.
4. Explain whether your estimate found in part (iii) is reliable.

1. A random sample of 10 cars of different makes and sizes is taken and the published miles per gallon, \(p\), and the actual miles per gallon, \(m\), are recorded. The data are coded using variables \(x = \frac { p } { 10 }\) and \(y = m - 25\)

The results for the coded data are summarised below. (You may use \(\sum y ^ { 2 } = 2628.25 \quad \sum x y = 768.58 \quad \mathrm {~S} _ { x x } = 9.25924 \quad \mathrm {~S} _ { x y } = 74.664\) )

1. Show that \(\mathrm { S } _ { y y } = 626.025\)
2. Find the product moment correlation coefficient between \(x\) and \(y\).
3. Give a reason to support fitting a regression model of the form \(y = a + b x\) to these data.
4. Find the equation of the regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\).
   Give the value of \(a\) and the value of \(b\) to 3 significant figures. A car's published miles per gallon is 44
5. Estimate the actual miles per gallon for this particular car.
6. Comment on the reliability of your estimate in part (e). Give a reason for your answer.

1. Two economics students, Andi and Behrouz, are studying some data relating to unemployment, \(x \%\), and increase in wages, \(y \%\), for a European country. The least squares regression line of \(y\) on \(x\) has equation

$$y = 3.684 - 0.3242 x$$ and $$\sum y = 23.7 \quad \sum y ^ { 2 } = 42.63 \quad \sum x ^ { 2 } = 756.81 \quad n = 16$$

1. Show that \(\mathrm { S } _ { y y } = 7.524375\)
2. Find \(\mathrm { S } _ { x x }\)
3. Find the product moment correlation coefficient between \(x\) and \(y\). Behrouz claims that, assuming the model is valid, the data show that when unemployment is 2\% wages increase at over 3\%
4. Explain how Behrouz could have come to this conclusion. Andi uses the formula $$\text { range } = \text { mean } \pm 3 \times \text { standard deviation }$$ to estimate the range of values for \(x\).
5. Find estimates of the minimum value and the maximum value of \(x\) in these data using Andi's formula.
6. Comment, giving a reason, on the reliability of Behrouz's claim. Andi suggests using the regression line with equation \(y = 3.684 - 0.3242 x\) to estimate unemployment when wages are increasing at \(2 \%\)
7. Comment, giving a reason, on Andi's suggestion.
   
   \includegraphics[max width=\textwidth, alt={}]{a439724e-b570-434d-bf75-de2b50915042-20_2647_1835_118_116}