Paired bivariate data $(x, y)$ is given in the table below. It represents the heights $x,m$ and lengths $y,m$ of a rare type of animal found on a small island.

$\begin{array}{|ccccccccc|}\hline x& 2.4& 3.6& 2.8& 1.8& 2.0& 2.2& 3.0& 3.4\\ y& 3.0& 4.0& 3.0& 1.7& 2.0& 2.3& 3.1& 2.7\\ \hline\end{array}$

Calculate the Pearson product moment correlation coefficient for these data.(Answer up to two decimal values)

Paired bivariate data $(x, y)$ is given in the table below. It represents the heights $x,m$ and lengths $y,m$ of a rare type of animal found on a small island.

$\begin{array}{|ccccccccc|}\hline x& 2.4& 3.6& 2.8& 1.8& 2.0& 2.2& 3.0& 3.4\\ y& 3.0& 4.0& 3.0& 1.7& 2.0& 2.3& 3.1& 2.7\\ \hline\end{array}$

State the type of linear correlation.

Paired bivariate data $(x, y)$ is given in the table below. It represents the heights $x,m$ and lengths $y,m$ of a rare type of animal found on a small island.

$\begin{array}{|ccccccccc|}\hline x& 2.4& 3.6& 2.8& 1.8& 2.0& 2.2& 3.0& 3.4\\ y& 3.0& 4.0& 3.0& 1.7& 2.0& 2.3& 3.1& 2.7\\ \hline\end{array}$

Calculate the $y$ on $x$ line of best fit.

Paired bivariate data $(x, y)$ is given in the table below. It represents the heights $x,m$ and lengths $y,m$ of a rare type of animal found on a small island.

$\begin{array}{|ccccccccc|}\hline x& 2.4& 3.6& 2.8& 1.8& 2.0& 2.2& 3.0& 3.4\\ y& 3.0& 4.0& 3.0& 1.7& 2.0& 2.3& 3.1& 2.7\\ \hline\end{array}$

Another four examples of this rare animal are found on a nearby smaller island. These extra data are given in the table below.

$\begin{array}{|lllll|}\hline x& 2.3& 2.7& 3.0& 3.5\\ y& 4.1& 1.5& 4.2& 1.5\\ \hline\end{array}$

Calculate the Pearson product moment correlation coefficient for the combined data of all twelve animals.(Answer up to three decimal values)

Paired bivariate data $(x, y)$ is given in the table below. It represents the heights $x,m$ and lengths $y,m$ of a rare type of animal found on a small island.

$\begin{array}{|ccccccccc|}\hline x& 2.4& 3.6& 2.8& 1.8& 2.0& 2.2& 3.0& 3.4\\ y& 3.0& 4.0& 3.0& 1.7& 2.0& 2.3& 3.1& 2.7\\ \hline\end{array}$

Another four examples of this rare animal are found on a nearby smaller island. These extra data are given in the table below.

$\begin{array}{|lllll|}\hline x& 2.3& 2.7& 3.0& 3.5\\ y& 4.1& 1.5& 4.2& 1.5\\ \hline\end{array}$

State the type of linear correlation that is shown in this example.

Paired bivariate data $(x, y)$ is given in the table below. It represents the heights $x,m$ and lengths $y,m$ of a rare type of animal found on a small island.

$\begin{array}{|ccccccccc|}\hline x& 2.4& 3.6& 2.8& 1.8& 2.0& 2.2& 3.0& 3.4\\ y& 3.0& 4.0& 3.0& 1.7& 2.0& 2.3& 3.1& 2.7\\ \hline\end{array}$

Another four examples of this rare animal are found on a nearby smaller island. These extra data are given in the table below.

$\begin{array}{|lllll|}\hline x& 2.3& 2.7& 3.0& 3.5\\ y& 4.1& 1.5& 4.2& 1.5\\ \hline\end{array}$

Suggest a reason why it would not be particularly valid to calculate the $y$ on $x$ line of best fit for the combined data.

 A set of bivariate data has a Pearson product moment correlation coefficient of $r=0.87$ for $25$ pairs $(x, y)$. The $y$ on $x$ line of best fit is given by $y=15x+11$. Consider the value of $r$ and line of best fit together with their definitions to answer the following. All the original $x$ -values are increased by adding $5$ and all the original $y$-values are decreased by subtracting $4$. State the new value of $r$.

A set of bivariate data has a Pearson product moment correlation coefficient of $r=0.87$ for $25$ pairs $(x, y)$. The $y$ on $x$ line of best fit is given by $y=15x+11$. Consider the value of $r$ and line of best fit together with their definitions to answer the following. All the original $x$ -values are increased by adding $5$ and all the original $y$-values are decreased by subtracting $4$. State the new value of $r$.

State the new value for the gradient of the $y$ on $x$ line of best fit.

 A set of bivariate data has a Pearson product moment correlation coefficient of $r=0.87$ for $25$ pairs $(x, y)$. The $y$ on $x$ line of best fit is given by $y=15x+11$. Consider the value of $r$ and line of best fit together with their definitions to answer the following. All the original $x$ -values are increased by adding $5$ and all the original $y$-values are decreased by subtracting $4$.

(i) State the new value of $r$.

(ii) State the new value for the gradient of the $y$ on $x$ line of best fit.

Give a reason for your answers to (i) iand (ii).