Estimate correlation from scatter diagram

1 The scatter diagrams below illustrate three sets of bivariate data, \(A , B\) and \(C\). \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} State, with an explanation in each case, which of the three sets of data has

1. the largest,
2. the smallest,
   value of the product moment correlation coefficient.

1. The scatter diagrams below were drawn by a student.

$$\begin{aligned} & y \underset { x } { \begin{array} { l l l l } & &
+ & & &
+ & + & + &
+ & + & + \end{array} } \end{aligned}$$ The student calculated the value of the product moment correlation coefficient for each of the sets of data. The values were $$\begin{array} { l l l } 0.68 & - 0.79 & 0.08 \end{array}$$ Write down, with a reason, which value corresponds to which scatter diagram.
(6)

1

1. Estimate, without undertaking any calculations, the value of the product moment correlation coefficient between the variables \(x\) and \(y\) for each of the two scatter diagrams. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{(i)} \includegraphics[alt={},max width=\textwidth]{156f9453-ebc6-4406-b5bc-08d1918ebc62-02_487_652_733_356}
   \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{156f9453-ebc6-4406-b5bc-08d1918ebc62-02_576_714_733_1153}
2. The table gives the circumference, \(x\) centimetres, and the weight, \(y\) grams, of each of 12 new cricket balls.

   \(\boldsymbol { x }\)	22.5	22.7	22.6	22.4	22.5	22.8	22.6	22.7	22.8	22.4	22.9	22.6
   \(\boldsymbol { y }\)	160.3	159.4	157.8	158.0	157.3	159.8	158.3	159.6	161.3	156.4	162.5	161.2

   1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
   2. Assuming that the 12 balls may be considered to be a random sample, interpret your value in context.

11 The scatter diagram shows data, taken from the pre-release data set, for several Local Authorities in one region of the UK in 2011. The diagram shows, for each Local Authority, the number of workers who drove to work, and the number of workers who walked to work. \begin{figure}[h] \end{figure}

1. Four students calculated the value of Pearson's product-moment correlation coefficient for the data in the diagram. Their answers were \(0.913,0.124 , - 0.913\) and - 0.124 . One of these values is correct. Without calculation state, with a reason, which is the correct value.
2. Sanjay makes the following statement.
   "The diagram shows that, in any Local Authority, if there are a large number of people who drive to work there will be a large number who walk to work." Give a reason why this statement is incorrect.
3. Rosie makes the following statement.
   "The diagram must be wrong because it shows good positive correlation. If there are more people driving to work, there will be fewer people walking to work, so there would be negative correlation." Explain briefly why Rosie's statement is incorrect.
4. The diagram shows a fairly close relationship between the two variables. One point on the diagram represents a Local Authority where this relationship is less strong than for the others. On the diagram in the Printed Answer Booklet, label this point A.
5. Given that the point A represents a metropolitan borough, suggest a reason why the relationship is less strong for this Local Authority than for the others in the region. The scatter diagram below shows the corresponding data for the same region in 2001. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{2001} \includegraphics[alt={},max width=\textwidth]{6a6316e4-7b2d-4533-988a-4863d79ce668-09_481_885_388_591}
   \end{figure}
6. (a) State a change that has taken place in the metropolitan borough represented by the point A between 2001 and 2011.
   (b) Suggest a possible reason for this change.

3 Estimate, without undertaking any calculations, the value of the product moment correlation coefficient between the variables \(x\) and \(y\) in each of the three scatter diagrams.

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_631_659_516_301}
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_620_647_525_1119}
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{868dc38b-3f24-4218-a300-c3cc2d9ff5d1-03_624_655_1279_303}
   (5 marks)