Find means from regression lines

A question is this type if and only if it asks to find the mean values of x and y given the equations of both regression lines (using the fact that both pass through the mean point).

5 questions · Moderate -0.1

5.08a Pearson correlation: calculate pmcc5.09c Calculate regression line
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OCR S1 2006 June Q1
6 marks Moderate -0.8
1 Some observations of bivariate data were made and the equations of the two regression lines were found to be as follows. $$\begin{array} { c c } y \text { on } x : & y = - 0.6 x + 13.0 \\ x \text { on } y : & x = - 1.6 y + 21.0 \end{array}$$
  1. State, with a reason, whether the correlation between \(x\) and \(y\) is negative or positive.
  2. Neither variable is controlled. Calculate an estimate of the value of \(x\) when \(y = 7.0\).
  3. Find the values of \(\bar { x }\) and \(\bar { y }\).
CAIE FP2 2008 November Q8
9 marks Moderate -0.3
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.
Edexcel S1 2023 October Q6
12 marks Moderate -0.3
  1. The variables \(x\) and \(y\) have the following regression equations based on the same 12 observations.
\cline { 2 - 2 } \multicolumn{1}{c|}{}Regression equation
\(y\) on \(x\)\(y = 1.4 x + 1.5\)
\(x\) on \(y\)\(x = 1.2 + 0.2 y\)
    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 \(y\)
CAIE FP2 2018 November Q9
11 marks Standard +0.8
For a random sample of 5 observations of pairs of values \((x, y)\), the equation of the regression line of \(y\) on \(x\) is \(y = -4.2 + c\) and the equation of the regression line of \(x\) on \(y\) is \(x = 10.8 + dy\), where \(c\) and \(d\) are constants. The product moment correlation coefficient is \(-0.7214\) and the mean value of \(x\) is 7.018. \begin{enumerate}[label=(\roman*)] \item Test at the 5% significance level whether there is evidence of non-zero correlation between the variables. [4] \item Find the values of \(c\) and \(d\). [5] \item Use an appropriate regression line to estimate the value of \(x\) when \(y = 3.5\), and comment on the reliability of your estimate. [2] \end{enumerate]
OCR S1 2010 January Q6
7 marks Standard +0.3
  1. A student calculated the values of the product moment correlation coefficient, \(r\), and Spearman's rank correlation coefficient, \(r_s\), for two sets of bivariate data, \(A\) and \(B\). His results are given below. $$A: \quad r = 0.9 \text{ and } r_s = 1$$ $$B: \quad r = 1 \quad \text{and } r_s = 0.9$$ With the aid of a diagram where appropriate, explain why the student's results for \(A\) could both be correct but his results for \(B\) cannot both be correct. [3]
  2. An old research paper has been partially destroyed. The surviving part of the paper contains the following incomplete information about some bivariate data from an experiment. \includegraphics{figure_6} The mean of \(x\) is 4.5. The equation of the regression line of \(y\) on \(x\) is \(y = 2.4x + 3.7\). The equation of the regression line of \(x\) on \(y\) is \(x = 0.40y\) + [missing constant] Calculate the missing constant at the end of the equation of the second regression line. [4]