Relate two regression lines

9 A set of 20 pairs of bivariate data \(( x , y )\) is summarised by $$\Sigma x = 200 , \quad \Sigma x ^ { 2 } = 2125 , \quad \Sigma y = 240 , \quad \Sigma y ^ { 2 } = 8245 .$$ The product moment correlation coefficient is - 0.992 .

1. What does the value of the product moment correlation coefficient indicate about a scatter diagram of the data points?
2. Find the equation of the regression line of \(y\) on \(x\).
3. The equation of the regression line of \(x\) on \(y\) is \(x = a ^ { \prime } + b ^ { \prime } y\). Find the value of \(b ^ { \prime }\).

For a random sample 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 $$y = - 0.5 x + 5 \quad \text { and } \quad x = - 1.2 y + 7.6$$ Find the value of the product moment correlation coefficient for this sample. Test, at the \(5 \%\) significance level, whether the population product moment correlation coefficient differs from zero. The following table shows the sample data. Find the values of \(p\) and \(q\).

9 A researcher records a random sample of \(n\) pairs of values of \(( x , y )\), giving the following summarised data. $$\Sigma x = 24 \quad \Sigma x ^ { 2 } = 160 \quad \Sigma y = 34 \quad \Sigma y ^ { 2 } = 324 \quad \Sigma x y = 192$$ The gradient of the regression line of \(y\) on \(x\) is \(- \frac { 3 } { 4 }\). Find

1. the value of \(n\),
2. the equation of the regression line of \(x\) on \(y\) in the form \(x = A y + B\), where \(A\) and \(B\) are constants to be determined,
3. the product moment correlation coefficient. Another researcher records the same data in the form \(\left( x ^ { \prime } , y ^ { \prime } \right)\), where \(x ^ { \prime } = \frac { x } { k } , y ^ { \prime } = \frac { y } { k }\) and \(k\) is a constant.
   Without further calculation, state the equation of the regression line of \(x ^ { \prime }\) on \(y ^ { \prime }\).

7 For a random sample of 10 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = 3.25 x - 4.27\). The sum of the ten \(x\) values is 15.6 and the product moment correlation coefficient for the sample is 0.56 . Find the equation of the regression line of \(x\) on \(y\). Test, at the \(5 \%\) significance level, whether there is evidence of non-zero correlation between the variables.

10 For a random sample of 6 observations of pairs of values \(( x , y )\), where \(0 < x < 21\) and \(0 < y < 14\), the following results are obtained. $$\Sigma x ^ { 2 } = 844.20 \quad \Sigma y ^ { 2 } = 481.50 \quad \Sigma x y = 625.59$$ It is also found that the variance of the \(x\)-values is 36.66 and the variance of the \(y\)-values is 9.69 .

1. Find the product moment correlation coefficient for the sample.
2. Find the equations of the regression lines of \(y\) on \(x\) and \(x\) on \(y\).
3. Use the appropriate regression line to estimate the value of \(x\) when \(y = 6.4\) and comment on the reliability of your estimate.

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 For a random sample of 10 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = 1.1664 + 0.4604 x\). It is given that $$\Sigma x ^ { 2 } = 1419.98 \quad \text { and } \quad \Sigma y ^ { 2 } = 439.68 .$$ The mean value of \(y\) is 6.24 .

1. Find the equation of the regression line of \(x\) on \(y\).
2. Find the product moment correlation coefficient.
3. Test at the \(5 \%\) significance level whether there is evidence of positive correlation between the two variables.

9 For a random sample of 5 observations of pairs of values \(( x , y )\), the equation of the regression line of \(y\) on \(x\) is \(y = 4.2 + c x\) and the equation of the regression line of \(x\) on \(y\) is \(x = 10.8 + d y\), where \(c\) and \(d\) are constants. The product moment correlation coefficient is - 0.7214 and the mean value of \(x\) is 7.018.

1. Test at the \(5 \%\) significance level whether there is evidence of non-zero correlation between the variables.
2. Find the values of \(c\) and \(d\).
3. Use an appropriate regression line to estimate the value of \(x\) when \(y = 3.5\), and comment on the reliability of your estimate.

1. Two variables \(x\) and \(y\) are such that, for a sample of ten pairs of values,

$$\sum x = 104 \cdot 5 , \quad \sum y = 113 \cdot 6 , \quad \sum x ^ { 2 } = 1954 \cdot 1 , \sum y ^ { 2 } = 2100 \cdot 6 .$$ The regression line of \(x\) on \(y\) has gradient 0.8 . Find

1. \(\sum x y\),
2. the equation of the regression line of \(y\) on \(x\),
3. the product moment correlation coefficient between \(y\) and \(x\).
4. Describe the kind of correlation indicated by your answer to (c).