5.08a Pearson correlation: calculate pmcc

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.

8 The yield of a particular crop on a farm is thought to depend principally on the amount of sunshine during the growing season. For a random sample of 8 years, the average yield, \(y\) kilograms per square metre, and the average amount of sunshine per day, \(x\) hours, are recorded. The results are given in the following table. $$\left[ \Sigma x = 72.4 , \Sigma x ^ { 2 } = 769.9 , \Sigma y = 78 , \Sigma y ^ { 2 } = 820 , \Sigma x y = 761.3 . \right]$$

1. Find the equation of the regression line of \(y\) on \(x\).
2. Find the product moment correlation coefficient.
3. Test, at the \(5 \%\) significance level, whether there is positive correlation between the average yield and the average amount of sunshine per day.

10 Delegates who travelled to a conference were asked to report the distance, \(y \mathrm {~km}\), that they had travelled and the time taken, \(x\) minutes. The values reported by a random sample of 8 delegates are given in the following table. $$\left[ \Sigma x = 610 , \Sigma x ^ { 2 } = 49682 , \Sigma y = 578 , \Sigma y ^ { 2 } = 45212 , \Sigma x y = 47136 . \right]$$ Find the equations of the regression lines of \(y\) on \(x\) and of \(x\) on \(y\). Estimate the time taken by a delegate who travelled 100 km to the conference. Calculate the product moment correlation coefficient for this sample.

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.

10 The lengths, \(x \mathrm {~m}\), and masses, \(y \mathrm {~kg}\), of 12 randomly chosen babies born at a particular hospital last year are summarised as follows. $$\Sigma x = 7.50 \quad \Sigma x ^ { 2 } = 4.73 \quad \Sigma y = 38.6 \quad \Sigma y ^ { 2 } = 124.84 \quad \Sigma x y = 24.25$$ Find the value of the product moment correlation coefficient for this sample. Obtain an estimate for the mass of a baby, born last year at the hospital, whose length is 0.64 m . Test, at the \(2 \%\) significance level, whether there is non-zero correlation between the two variables.

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 \(N\).

For a random sample, \(A\), of 5 pairs of values of \(x\) and \(y\), the equations of the regression lines of \(y\) on \(x\) and \(x\) on \(y\) are respectively \(y = 4.5 + 0.3 x\) and \(x = 3 y - 13\). Four of the five pairs of data are given in the following table. Find

1. the fifth pair of values of \(x\) and \(y\),
2. the value of the product moment correlation coefficient. A second random sample, \(B\), of 5 pairs of values of \(x\) and \(y\) is summarised as follows. $$\Sigma x = 20 \quad \Sigma x ^ { 2 } = 100 \quad \Sigma y = 17 \quad \Sigma y ^ { 2 } = 69 \quad \Sigma x y = 75$$ The two samples, \(A\) and \(B\), are combined to form a single random sample of size 10 .
3. Use this combined sample to test, at the \(5 \%\) significance level, whether the population product moment correlation coefficient is different from zero.

9 The land areas \(x\) (in suitable units) and populations \(y\) (in millions) for a sample of 8 randomly chosen cities are given in the following table. $$\left[ \Sigma x = 35.9 , \Sigma x ^ { 2 } = 216.47 , \Sigma y = 36.8 , \Sigma y ^ { 2 } = 244.96 , \Sigma x y = 212.62 . \right]$$

1. Find, showing all necessary working, the value of the product moment correlation coefficient for this sample.
2. Using a \(1 \%\) significance level, test whether there is positive correlation between land area and population of cities.
   The land areas and populations for another randomly chosen sample of cities, this time of size \(n\), give a product moment correlation coefficient of 0.651 . Using a test at the \(1 \%\) significance level, there is evidence of non-zero correlation between the variables.
3. Find the least possible value of \(n\), justifying your answer.

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.

11 Christa used Pearson's product-moment correlation coefficient, \(r\), to compare the use of public transport with the use of private vehicles for travel to work in the UK.

1. Using the pre-release data set for all 348 UK Local Authorities, she considered the following four variables.

   Number of employees using public transport	\(x\)
   Number of employees using private vehicles	\(y\)
   Proportion of employees using public transport	\(a\)
   Proportion of employees using private vehicles	\(b\)

   1. Explain, in context, why you would expect strong, positive correlation between \(x\) and \(y\).
   2. Explain, in context, what kind of correlation you would expect between \(a\) and \(b\).
   3. Christa also considered the data for the 33 London boroughs alone and she generated the following scatter diagram. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{London} \includegraphics[alt={},max width=\textwidth]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-07_467_707_1366_653}
      \end{figure} One London Borough is represented by an outlier in the diagram.
      (a) Suggest what effect this outlier is likely to have on the value of \(r\) for the 32 London Boroughs.
      (b) Suggest what effect this outlier is likely to have on the value of \(r\) for the whole country.
   4. What can you deduce about the area of the London Borough represented by the outlier? Explain your answer.

11 A trainer was asked to give a lecture on population profiles in different Local Authorities (LAs) in the UK. Using data from the 2011 census, he created the following scatter diagram for 17 selected LAs. \begin{figure}[h] \end{figure} He selected the 17 LAs using the following method. The proportions of people aged 18 to 24 and aged 65+ in any Local Authority are denoted by \(P _ { \text {young } }\) and \(P _ { \text {senior } }\) respectively. The trainer used a spreadsheet to calculate the value of \(k = \frac { P _ { \text {young } } } { P _ { \text {senior } } }\) for each of the 348 LAs in the UK. He then used specific ranges of values of \(k\) to select the 17 LAs.

1. Estimate the ranges of values of \(k\) that he used to select these 17 LAs.
2. Using the 17 LAs the trainer carried out a hypothesis test with the following hypotheses. \(\mathrm { H } _ { 0 }\) : There is no linear correlation in the population between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\). \(\mathrm { H } _ { 1 }\) : There is negative linear correlation in the population between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\).
   He found that the value of Pearson's product-moment correlation coefficient for the 17 LAs is - 0.797 , correct to 3 significant figures.
   1. Use the table on page 9 to show that this value is significant at the \(1 \%\) level. The trainer concluded that there is evidence of negative linear correlation between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\) in the population.
   2. Use the diagram to comment on the reliability of this conclusion.
3. Describe one outstanding feature of the population in the areas represented by the points in the bottom right hand corner of the diagram.
4. The trainer's audience included representatives from several universities. Suggest a reason why the diagram might be of particular interest to these people. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Critical values of Pearson's product-moment correlation coefficient}

   \multirow{2}{*}{1-tail test 2-tail test}	5\%	2.5\%	1\%	0.5\%
   	10\%	5\%	2\%	1\%
   \(n\)
   1	-	-	-	-
   2	-	-	-	-
   3	0.9877	0.9969	0.9995	0.9999
   4	0.9000	0.9500	0.9800	0.9900
   5	0.8054	0.8783	0.9343	0.9587
   6	0.7293	0.8114	0.8822	0.9172
   7	0.6694	0.7545	0.8329	0.8745
   8	0.6215	0.7067	0.7887	0.8343
   9	0.5822	0.6664	0.7498	0.7977
   10	0.5494	0.6319	0.7155	0.7646
   11	0.5214	0.6021	0.6851	0.7348
   12	0.4973	0.5760	0.6581	0.7079
   13	0.4762	0.5529	0.6339	0.6835
   14	0.4575	0.5324	0.6120	0.6614
   15	0.4409	0.5140	0.5923	0.6411
   16	0.4259	0.4973	0.5742	0.6226
   17	0.4124	0.4821	0.5577	0.6055
   18	0.4000	0.4683	0.5425	0.5897
   19	0.3887	0.4555	0.5285	0.5751
   20	0.3783	0.4438	0.5155	0.5614
   21	0.3687	0.4329	0.5034	0.5487
   22	0.3598	0.4227	0.4921	0.5368
   23	0.3515	0.4132	0.4815	0.5256
   24	0.3438	0.4044	0.4716	0.5151
   25	0.3365	0.3961	0.4622	0.5052
   26	0.3297	0.3882	0.4534	0.4958
   27	0.3233	0.3809	0.4451	0.4869
   28	0.3172	0.3739	0.4372	0.4785
   29	0.3115	0.3673	0.4297	0.4705
   30	0.3061	0.3610	0.4226	0.4629

   \end{table} Turn over for questions 12 and 13

10 A researcher plans to carry out a statistical investigation to test whether there is linear correlation between the time ( \(T\) weeks) from conception to birth, and the birth weight ( \(W\) grams) of new-born babies.

1. Explain why a 1-tail test is appropriate in this context. The researcher records the values of \(T\) and \(W\) for a random sample of 11 babies. They calculate Pearson's product-moment correlation coefficient for the sample and find that the value is 0.722 .
2. Use the table below to carry out the test at the \(1 \%\) significance level. \section*{Critical values of Pearson's product-moment correlation coefficient.}

   \multirow{2}{*}{}	1-tail test	5\%	2.5\%	1\%	0.5\%
   	2-tail test	10\%	5\%	2.5\%	1\%
   \multirow{4}{*}{\(n\)}	10	0.5494	0.6319	0.7155	0.7646
   	11	0.5214	0.6021	0.6851	0.7348
   	12	0.4973	0.5760	0.6581	0.7079
   	13	0.4762	0.5529	0.6339	0.6835

14 The pre-release material contains information concerning the median income of taxpayers in \(\pounds\) and the percentage of all pupils at the end of KS4 achieving 5 or more GCSEs at grade A*-C, including English and Maths, for different areas of London. Some of the data for 2014/15 is shown in Fig. 14.1. \begin{table}[h] \end{table} A student investigated whether there is any relationship between median income of taxpayers and percentage of pupils achieving 5 or more GCSEs at grade A*-C, including English and Maths.

1. With reference to Fig. 14.1, explain how the data should be cleaned before any analysis can take place. After the data was cleaned, the student used software to draw the scatter diagram shown in Fig. 14.2. Scatter diagram to show percentage of pupils achieving 5 A*-C grades against median income of taxpayers \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 14.2} \includegraphics[alt={},max width=\textwidth]{11788aaf-98fb-4a78-8a40-a40743b1fe15-10_574_1481_1900_241}
   \end{figure} The student calculated that the product moment correlation coefficient for these data is 0.3743 .
2. Give two reasons why it may not be appropriate to use a linear model for the relationship between median income of taxpayers in \(\pounds\) and the percentage of all pupils at the end of KS4 achieving 5 or more GCSEs at grade A*-C. The student carried out some further analysis. The results are shown in Fig. 14.3. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 14.3}

   	median income of taxpayers in \(\pounds\)	percentage of pupils achieving \(5 + \mathrm { A } ^ { * } - \mathrm { C }\)
   mean	27216	61.0
   standard deviation	4177.5	5.32

   \end{table} The student identified three outliers in total.
3. The student decided to remove these outliers and recalculate the product moment correlation coefficient.
4. Explain whether the new value of the product moment correlation coefficient would be between 0.3743 and 1 or between 0 and 0.3743 .

4 Judith believes that mathematical ability and chess-playing ability are related. She asks 20 randomly chosen chess players, with known British Chess Federation (BCF) ratings \(X\), to take a mathematics aptitude test, with scores \(Y\). The results are summarised as follows. $$n = 20 , \sum x = 3600 , \sum x ^ { 2 } = 660500 , \sum y = 1440 , \sum y ^ { 2 } = 105280 , \sum x y = 260990$$

1. Calculate the value of Pearson's product-moment correlation coefficient \(r\).
2. State an assumption needed to be able to carry out a significance test on the value of \(r\).
3. Assume now that the assumption in part (ii) is valid. Test at the \(5 \%\) significance level whether there is evidence that chess players with higher BCF ratings are better at mathematics.
4. There are two different grading systems for chess players, the BCF system and the international ELO system. The two sets of ratings are related by $$\text { ELO rating } = 8 \times \text { BCF rating } + 650$$ Magnus says that the experiment should have used ELO ratings instead of BCF ratings. Comment on Magnus's suggestion.

5 Sixteen candidates took an examination paper in mechanics and an examination paper in statistics.

1. For all sixteen candidates, the value of the product moment correlation coefficient \(r\) for the marks on the two papers was 0.701 correct to 3 significant figures. Test whether there is evidence, at the \(5 \%\) significance level, of association between the marks on the two papers.
2. A teacher decided to omit the marks of the candidates who were in the top three places in mechanics and the candidates who were in the bottom three places in mechanics. The marks for the remaining 10 candidates can be summarised by \(n = 10 , \sum x = 750 , \sum y = 690 , \sum x ^ { 2 } = 57690 , \sum y ^ { 2 } = 49676 , \sum x y = 50829\).
   1. Calculate the value of \(r\) for these 10 candidates.
   2. What do the two values of \(r\), in parts (a) and (b)(i), tell you about the scores of the sixteen candidates?

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.

5 A psychologist investigates the relationship between 'openness' and 'creativity' in adults. Each member of a random sample of 15 adults is given two tests, one on openness and one on creativity. Each test has a maximum score of 75 . The results are given in the table. \(n = 15 \quad \sum x = 519 \quad \sum y = 645 \quad \sum x ^ { 2 } = 19033 \quad \sum y ^ { 2 } = 29751 \quad \sum x y = 23034\)

1. Use Pearson's product-moment correlation coefficient to test, at the \(5 \%\) significance level, whether there is positive association between openness and creativity.
2. State what the value of Pearson's product-moment correlation coefficient shows about a scatter diagram illustrating the data.
3. A student suggests that there is a way to obtain a more accurate measure of the correlation. Before carrying out the test it would be better to standardise the test scores so that they have the same mean and variance. Explain whether you agree with this suggestion.

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).

1 Five observations of bivariate data \(( x , y )\) are given in the table.

1. Find the value of Pearson's product-moment correlation coefficient.
2. State what your answer to part (a) tells you about a scatter diagram representing the data.
3. A new variable \(a\) is defined by \(\mathrm { a } = 3 \mathrm { x } + 4\). Dee says "The value of Pearson's product-moment correlation coefficient between \(a\) and \(y\) will not be the same as the answer to part (a)." State with a reason whether you agree with Dee.

2 A shopper estimates the cost, \(\pounds X\) per item, of each of 12 items in a supermarket. The shopper's estimates are compared with the actual cost, \(\pounds Y\) per item, of each item. The results are summarised as follows. \(n = 12\) \(\sum x = 399\) \(\sum y = 623.88\) \(\sum x ^ { 2 } = 28127\) \(\sum y ^ { 2 } = 116509.0212\) \(\sum x y = 45006.01\) Test at the 1\% significance level whether the shopper's estimates are positively correlated with the actual cost of the items.

1 A set of bivariate data ( \(X , Y\) ) is summarised as follows. \(n = 25 , \sum x = 9.975 , \sum y = 11.175 , \sum x ^ { 2 } = 5.725 , \sum y ^ { 2 } = 46.200 , \sum x y = 11.575\)

1. Calculate the value of Pearson's product-moment correlation coefficient.
2. Calculate the equation of the regression line of \(y\) on \(x\). It is desired to know whether the regression line of \(y\) on \(x\) will provide a reliable estimate of \(y\) when \(x = 0.75\).
3. State one reason for believing that the estimate will be reliable.
4. State what further information is needed in order to determine whether the estimate is reliable.

2 The directors of a large company believe that there are more computer failures in the Head Office when temperatures are higher. They obtain data for the Head Office for the maximum temperature, \(T ^ { \circ } \mathrm { C }\), and the number of computer failures, \(X\), on each of 12 randomly chosen days.

1. State which of the following words can be applied to \(T\). Dependent Independent Controlled Response The data is summarised as follows. \(n = 12 \quad \sum t = 261 \quad \sum x = 41 \quad \sum t ^ { 2 } = 5869 \quad \sum x ^ { 2 } = 311 \quad \sum \mathrm { tx } = 1021\)
2. Calculate the value of the product moment correlation coefficient \(r\).
3. The directors wish to investigate their belief using a significance test at the \(1 \%\) level.
   1. Explain why a 1-tail test is appropriate in this situation.
   2. Carry out the test.
4. One of the directors prefers the temperatures to be given in Fahrenheit ( \({ } ^ { \circ } \mathrm { F }\) ), rather than Centigrade ( \({ } ^ { \circ } \mathrm { C }\) ). The relationship between F and C is \(\mathrm { F } = \frac { 9 } { 5 } \mathrm { C } + 32\).
   State the value of \(r\) that would result from using temperatures in Fahrenheit in the calculation.

2 A newspaper article claimed that "taller dog owners have taller dogs as pets". Alex investigated this claim and obtained data from a random sample of 16 fellow students who owned exactly one dog. The results are summarised as follows, where the height of the student, in cm, is denoted by \(h\) and the height, in cm, of their dog is denoted by \(d\). \(\mathrm { n } = 16 \quad \sum \mathrm {~h} = 2880 \quad \sum \mathrm {~d} = 660 \quad \sum \mathrm {~h} ^ { 2 } = 519276 \quad \sum \mathrm {~d} ^ { 2 } = 30000 \quad \sum \mathrm { hd } = 119425\)

1. Calculate the value of Pearson's product moment correlation coefficient for the data.
2. State what your answer tells you about a scatter diagram illustrating the data.
3. Use the data to test, at the \(5 \%\) significance level, the claim of the newspaper article.
4. Explain whether the answer to part (a) would be likely to be different if the dogs' weights had been used instead of their heights.

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. 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.