2.02e Correlation does not imply causation

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1. The table shows the amount, \(x\), in hundreds of pounds, spent on heating and the number of absences, \(y\), at a factory during each month in 2014.

   Amount, \(x\), spent on heating (£ hundreds)	21	23	19	15	14	5	2	10	9	20	18	23
   Number of absences, \(y\)	23	25	18	18	12	10	4	9	11	15	20	26

   \(n = 12 \quad \Sigma x = 179 \quad \Sigma x ^ { 2 } = 3215 \quad \Sigma y = 191 \quad \Sigma y ^ { 2 } = 3565 \quad \Sigma x y = 3343\)
   1. Calculate \(r\), the product moment correlation coefficient, showing that \(r > 0.92\).
   2. A manager says, 'The value of \(r\) shows that spending more money on heating causes more absences, so we should spend less on heating.' Comment on this claim.
   3. The months in 2014 were numbered \(1,2,3 , \ldots , 12\). The output, \(z\), in suitable units was recorded along with the month number, \(n\), for each month in 2014. The equation of the regression line of \(z\) on \(n\) was found to be \(z = 0.6 n + 17\).
      (a) Use this equation to explain whether output generally increased or decreased over these months.
      (b) Find the mean of \(n\) and use the equation of the regression line to calculate the mean of \(z\).
   4. Hence calculate the total output in 2014.

1 For each of the last five years the number of tourists, \(x\) thousands, visiting Sackton, and the average weekly sales, \(\pounds y\) thousands, in Sackton Stores were noted. The table shows the results.

1. Calculate the product moment correlation coefficient \(r\) between \(x\) and \(y\).
2. It is required to estimate the average weekly sales at Sackton Stores in a year when the number of tourists is 280000 . Calculate the equation of an appropriate regression line, and use it to find this estimate.
3. Over a longer period the value of \(r\) is - 0.8 . The mayor says, "This shows that having more tourists causes sales at Sackton Stores to decrease." Give a reason why this statement is not correct.

11 A student is investigating changes in the number of residents in Local Authorities in the SouthEast Region between 2001 and 2011. The scatter diagram shows the number \(x\) of residents in these Local Authorities in the age group 8 to 9 in 2001 and the number \(y\) of residents in the same Local Authorities in the age group 18 to 19 in 2011.

1. Suggest a reason why the student is comparing these two age groups in 2001 and 2011. The student notices that most of the data points are close to the line \(y = x\).
2. 1. Explain what this suggests about the residents in these Local Authorities.
   2. The student says that correlation does not imply causation, so there is no causal link between the values of \(x\) and the values of \(y\). Explain whether or not they are correct.
3. Some of these Local Authorities contain universities.
   1. On the diagram in the Printed Answer Booklet, circle three points that are likely to represent Local Authorities containing universities.
   2. Give a reason for your choice of points in part (c)(i). Assume that the proportion of residents in age group 8 to 9 in 2001 was roughly the same in each Local Authority in the South-East. The Local Authority in this region with the largest population is Medway.
4. On the diagram in the Printed Answer Booklet, label clearly with the letter \(M\) the point that corresponds to Medway.

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The number of hours of sunshine and the daily maximum temperature were recorded over a 9-day period in June at an English seaside town. A scatter diagram representing the recorded data is shown below. \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-20_872_1511_488_264} One of the points on the scatter diagram is an error. 15

1. 1. Write down the letter that identifies this point.
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      1. (ii) Suggest one possible action that could be taken to deal with this error.
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   2. It is claimed that the scatter diagram proves that longer hours of sunshine cause
      higher maximum daily temperatures. Comment on the validity of this claim.
      [0pt] [1 mark]

This question deals with information about the populations of Local Authorities (LAs) in the North of England, taken from the 2011 census. \includegraphics{figure_6} Fig. 1 and Fig. 2 both show strong correlation, but of two different kinds.

1. For each diagram, use a single word to describe the kind of correlation shown. [1]
2. For each diagram, suggest a reason, in context, why the correlation is of the particular kind described in part (a). [2]

Fig. 3 is the same as Fig. 2 but with the point \(A\) marked. Fig. 4 shows information about the same LAs as Fig. 2 and Fig. 3. \includegraphics{figure_7}

1. Point \(A\) in Fig. 3 and point \(B\) in Fig. 4 represent the same LA. Explain how you can tell that this LA has a large population. [1]

The following table gives the mean per capita consumption of mozzarella cheese per annum, \(x\) pounds, and the number of civil engineering doctorates awarded, \(y\), in the United States in each of 10 years.

\(x\)	9.3	9.7	9.7	9.7	9.9	10.2	10.5	11.0	10.6	10.6
\(y\)	480	501	540	552	547	622	655	701	712	708

source: www.tylervigen.com

1. Find the equation of the regression line of \(y\) on \(x\). [2]

You are given that the product moment correlation coefficient is 0.959.

1. Explain whether this value would be different if \(x\) is measured in kilograms instead of pounds. [1]

It is desired to carry out a hypothesis test to investigate whether there is correlation between these two variables.

1. Assume that the data is a random sample of all years.
   1. Carry out the test at the 10\% significance level. [6]
   2. Explain whether your conclusion suggests that manufacturers of mozzarella cheese could increase consumption by sponsoring doctoral candidates in civil engineering. [1]