5.08a Pearson correlation: calculate pmcc

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. Ranpose hospital offers services to a large number of clinics that refer patients to a range of hospitals.
   The manager at Ranpose hospital took a random sample of 16 clinics and recorded

• the distance, \(x \mathrm {~km}\), of the clinic from Ranpose hospital
• the percentage, \(y \%\), of the referrals from the clinic who attend Ranpose hospital.

The data are summarised as $$\bar { x } = 8.1 \quad \bar { y } = 20.5 \quad \sum y ^ { 2 } = 8266 \quad \mathrm {~S} _ { x x } = 368.16 \quad \mathrm {~S} _ { x y } = - 630.9$$

1. Find the product moment correlation coefficient for these data.
2. Give an interpretation of your correlation coefficient. The manager at Ranpose hospital believes that there may be a linear relationship between the distance of a clinic from the hospital and the percentage of the referrals who attend the hospital. She drew the following scatter diagram for these data. \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-20_1106_926_1133_511}
3. State, giving a reason, whether or not these data support the manager's belief.
   (1)
   \section*{[The summary data and the scatter diagram are repeated below.]} The data are summarised as $$\bar { x } = 8.1 \quad \bar { y } = 20.5 \quad \sum y ^ { 2 } = 8266 \quad \mathrm {~S} _ { x x } = 368.16 \quad \mathrm {~S} _ { x y } = - 630.9$$ \includegraphics[max width=\textwidth, alt={}, center]{9ac7647f-b291-4a64-9518-fa6438a0cc7d-22_1118_936_612_504}
4. Find the equation of the regression line of \(y\) on \(x\), giving your answer in the form $$y = a + b x$$
5. Give an interpretation of the gradient of your regression line.
6. Draw your regression line on the scatter diagram. The manager believes that Ranpose hospital should be attracting an "above average" percentage of referrals from clinics that are less than 5 km from the hospital. She proposes to target one clinic with some extra publicity about the services Ranpose offers.
7. On the scatter diagram circle the point representing the clinic she should target.

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1. A large company rents shops in different parts of the country. A random sample of 10 shops was taken and the floor area, \(x\) in \(10 \mathrm {~m} ^ { 2 }\), and the annual rent, \(y\) in thousands of dollars, were recorded.
   The data are summarised by the following statistics

$$\sum x = 900 \quad \sum x ^ { 2 } = 84818 \quad \sum y = 183 \quad \sum y ^ { 2 } = 3434$$ and the regression line of \(y\) on \(x\) has equation \(y = 6.066 + 0.136 x\)

1. Use the regression line to estimate the annual rent in dollars for a shop with a floor area of \(800 \mathrm {~m} ^ { 2 }\)
2. Find \(\mathrm { S } _ { y y }\) and \(\mathrm { S } _ { x x }\)
3. Find the product moment correlation coefficient between \(y\) and \(x\). An 11th shop is added to the sample. The floor area is \(900 \mathrm {~m} ^ { 2 }\) and the annual rent is 15000 dollars.
4. Use the formula \(\mathrm { S } _ { x y } = \sum ( x - \bar { x } ) ( y - \bar { y } )\) to show that the value of \(\mathrm { S } _ { x y }\) for the 11 shops will be the same as it was for the original 10 shops.
5. Find the new equation of the regression line of \(y\) on \(x\) for the 11 shops. The company is considering renting a larger shop with area of \(3000 \mathrm {~m} ^ { 2 }\)
6. Comment on the suitability of using the new regression line to estimate the annual rent. 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}

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. A doctor is studying the scans of 30 -week old foetuses. She takes a random sample of 8 scans and measures the length, \(f \mathrm {~mm}\), of the leg bone called the femur. She obtains the following results.

$$\begin{array} { l l l l l l l l } 52 & 53 & 56 & 57 & 57 & 59 & 60 & 62 \end{array}$$

1. Show that \(\mathrm { S } _ { f f } = 80\) The doctor also measures the head circumference, \(h \mathrm {~mm}\), of each foetus and her results are summarised as $$\sum h = 2209 \quad \sum h ^ { 2 } = 610463 \quad \mathrm {~S} _ { f h } = 182$$
2. Find \(\mathrm { S } _ { h h }\)
3. Calculate the product moment correlation coefficient between the length of the femur and the head circumference for these data. The doctor believes that there is a linear relationship between the length of the femur and the head circumference of 30-week old foetuses.
4. State, giving a reason, whether or not your calculation in part (c) supports the doctor's belief.
5. Find an equation of the regression line of \(h\) on \(f\). The doctor plans in future to measure the femur length, \(f\), and then use the regression line to estimate the corresponding head circumference, \(h\). A statistician points out that there will always be the chance of an error between the true head circumference and the estimated value of the head circumference. Given that the error, \(E \mathrm {~mm}\), has the normal distribution \(\mathrm { N } \left( 0,4 ^ { 2 } \right)\)
6. find the probability that the estimate is within 3 mm of the true value.

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 variables \(x\) and \(y\) have the following regression equations based on the same 12 observations.

1. 1. Find the point of intersection of these lines.
   2. Hence show that \(\sum x = 25\) Given that $$\sum x y = \frac { 6961 } { 60 }$$
2. Find \(S _ { x y }\)
3. Find the product moment correlation coefficient between \(x\) and \(y\)

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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1. Gary compared the total attendance, \(x\), at home matches and the total number of goals, \(y\), scored at home during a season for each of 12 football teams playing in a league. He correctly calculated:

$$S _ { x x } = 1022500 \quad S _ { y y } = 130.9 \quad S _ { x y } = 8825$$

1. Calculate the product moment correlation coefficient for these data.
2. Interpret the value of the correlation coefficient. Helen was given the same data to analyse. In view of the large numbers involved she decided to divide the attendance figures by 100 . She then calculated the product moment correlation coefficient between \(\frac { x } { 100 }\) and \(y\).
3. Write down the value Helen should have obtained.

6. A local authority is investigating the cost of reconditioning its incinerators. Data from 10 randomly chosen incinerators were collected. The variables monitored were the operating time \(x\) (in thousands of hours) since last reconditioning and the reconditioning cost \(y\) (in \(\pounds 1000\) ). None of the incinerators had been used for more than 3000 hours since last reconditioning. The data are summarised below, $$\Sigma x = 25.0 , \Sigma x ^ { 2 } = 65.68 , \Sigma y = 50.0 , \Sigma y ^ { 2 } = 260.48 , \Sigma x y = 130.64 .$$

1. Find \(\mathrm { S } _ { x x } , \mathrm {~S} _ { x y } , \mathrm {~S} _ { y y }\).
2. Calculate the product moment correlation coefficient between \(x\) and \(y\).
3. Explain why this value might support the fitting of a linear regression model of the form \(y = a + b x\).
4. Find the values of \(a\) and \(b\).
5. Give an interpretation of \(a\).
6. Estimate
   1. the reconditioning cost for an operating time of 2400 hours,
   2. the financial effect of an increase of 1500 hours in operating time.
7. Suggest why the authority might be cautious about making a prediction of the reconditioning cost of an incinerator which had been operating for 4500 hours since its last reconditioning.

6. The chief executive of Rex cars wants to investigate the relationship between the number of new car sales and the amount of money spent on advertising. She collects data from company records on the number of new car sales, \(c\), and the cost of advertising each year, \(p\) (£000). The data are shown in the table below.

1. Using the coding \(x = ( p - 100 )\) and \(y = \frac { 1 } { 10 } ( c - 4000 )\), draw a scatter diagram to represent these data. Explain why \(x\) is the explanatory variable.
2. Find the equation of the least squares regression line of \(y\) on \(x\). $$\text { [Use } \left. \Sigma x = 402 , \Sigma y = 517 , \Sigma x ^ { 2 } = 17538 \text { and } \Sigma x y = 22611 . \right]$$
3. Deduce the equation of the least squares regression line of \(c\) on \(p\) in the form \(c = a + b p\).
4. Interpret the value of \(a\).
5. Predict the number of extra new cars sales for an increase of \(\pounds 2000\) in advertising budget. Comment on the validity of your answer.
   (2)

3. The following table shows the height \(x\), to the nearest cm , and the weight \(y\), to the nearest kg , of a random sample of 12 students.

1. On graph paper, draw a scatter diagram to represent these data.
2. Write down, with a reason, whether the correlation coefficient between \(x\) and \(y\) is positive or negative. The data in the table can be summarised as follows. $$\Sigma x = 1962 , \quad \Sigma y = 740 , \quad \Sigma y ^ { 2 } = 47746 , \quad \Sigma x y = 122783 , \quad S _ { x x } = 1745 .$$
3. Find \(S _ { x y }\). The equation of the regression line of \(y\) on \(x\) is \(y = - 106.331 + b x\).
4. Find, to 3 decimal places, the value of \(b\).
5. Find, to 3 significant figures, the mean \(\bar { y }\) and the standard deviation \(s\) of the weights of this sample of students.
6. Find the values of \(\bar { y } \pm 1.96 s\).
7. Comment on whether or not you think that the weights of these students could be modelled by a normal distribution.

1. As part of a statistics project, Gill collected data relating to the length of time, to the nearest minute, spent by shoppers in a supermarket and the amount of money they spent. Her data for a random sample of 10 shoppers are summarised in the table below, where \(t\) represents time and \(\pounds m\) the amount spent over \(\pounds 20\).

1. Write down the actual amount spent by the shopper who was in the supermarket for 15 minutes.
2. Calculate \(S _ { t t } , S _ { m m }\) and \(S _ { t m }\). $$\text { (You may use } \Sigma t ^ { 2 } = 5478 \Sigma m ^ { 2 } = 2101 \Sigma t m = 2485 \text { ) }$$
3. Calculate the value of the product moment correlation coefficient between \(t\) and \(m\).
4. Write down the value of the product moment correlation coefficient between \(t\) and the actual amount spent. Give a reason to justify your value. On another day Gill collected similar data. For these data the product moment correlation coefficient was 0.178
5. Give an interpretation to both of these coefficients.
6. Suggest a practical reason why these two values are so different.

1. A personnel manager wants to find out if a test carried out during an employee's interview and a skills assessment at the end of basic training is a guide to performance after working for the company for one year.

The table below shows the results of the interview test of 10 employees and their performance after one year. $$\text { [You may use } \sum x ^ { 2 } = 60475 , \sum y ^ { 2 } = 53122 , \sum x y = 56076 \text { ] }$$

1. Showing your working clearly, calculate the product moment correlation coefficient between the interview test and the performance after one year. The product moment correlation coefficient between the skills assessment and the performance after one year is - 0.156 to 3 significant figures.
2. Use your answer to part (a) to comment on whether or not the interview test and skills assessment are a guide to the performance after one year. Give clear reasons for your answers.

4. In a study of how students use their mobile telephones, the phone usage of a random sample of 11 students was examined for a particular week. The total length of calls, \(y\) minutes, for the 11 students were $$17,23,35,36,51,53,54,55,60,77,110$$

1. Find the median and quartiles for these data. A value that is greater than \(Q _ { 3 } + 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) or smaller than \(Q _ { 1 } - 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right)\) is defined as an outlier.
2. Show that 110 is the only outlier.
3. Using the graph paper on page 15 draw a box plot for these data indicating clearly the position of the outlier. The value of 110 is omitted.
4. Show that \(S _ { y y }\) for the remaining 10 students is 2966.9 These 10 students were each asked how many text messages, \(x\), they sent in the same week. The values of \(S _ { x x }\) and \(S _ { x y }\) for these 10 students are \(S _ { x x } = 3463.6\) and \(S _ { x y } = - 18.3\).
5. Calculate the product moment correlation coefficient between the number of text messages sent and the total length of calls for these 10 students. A parent believes that a student who sends a large number of text messages will spend fewer minutes on calls.
6. Comment on this belief in the light of your calculation in part (e). \includegraphics[max width=\textwidth, alt={}, center]{d5d000c7-de42-461a-ba05-6c8b2c333780-09_611_1593_297_178}

1. A random sample of 50 salmon was caught by a scientist. He recorded the length \(l \mathrm {~cm}\) and weight \(w \mathrm {~kg}\) of each salmon.

The following summary statistics were calculated from these data. \(\sum l = 4027 \quad \sum l ^ { 2 } = 327754.5 \quad \sum w = 357.1 \quad \sum l w = 29330.5 \quad S _ { w w } = 289.6\)

1. Find \(S _ { l l }\) and \(S _ { l w }\)
2. Calculate, to 3 significant figures, the product moment correlation coefficient between \(l\) and \(w\).
3. Give an interpretation of your coefficient.

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 young family were looking for a new 3 bedroom semi-detached house. A local survey recorded the price \(x\), in \(\pounds 1000\), and the distance \(y\), in miles, from the station of such houses. The following summary statistics were provided

$$S _ { x x } = 113573 , \quad S _ { y y } = 8.657 , \quad S _ { x y } = - 808.917$$

1. Use these values to calculate the product moment correlation coefficient.
2. Give an interpretation of your answer to part (a). Another family asked for the distances to be measured in km rather than miles.
3. State the value of the product moment correlation coefficient in this case.

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 \(19 ^ { \circ } \mathrm { C }\).