Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: Modelling Relationships between Two Data Sets: Statistics for Bivariate Data, Exercise 24: Chapter review

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 type of linear correlation that is shown in this example.

 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 $y$ -values are altered by multiplying by $2$ and all the original $x$ values remain unchanged.

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 $y$ -values are altered by multiplying by $2$ and all the original $x$ values remain unchanged.

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 $y$ -values are altered by multiplying by $2$ and all the original $x$ values remain unchanged.

(i) State the new value of $r$.

(ii) 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 altered by multiplying by $-2$ and all the original $y$ -values remain unchanged.

State the new value of $r$.

The line of best fit coefficients are independent of the change of the origin. But, they are not independent of the change of the scale. It means there will be no effect on the line of best fit coefficients if any constant is subtracted from the value of $x$ and $y$.

From the regression line $y=ax+b$, $a$ represents gradient and $b$ represents $y$-intercept.The gradient is $-2a$ if the $y$-values are altered by multiplying by $-2$.

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 altered by multiplying by $-2$ and all the original $y$ -values remain unchanged.

(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) and (ii).

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 altered by multiplying by $-2$ and all the original $y$ -values remain unchanged.

Describe in two words the linear correlation that exists for the new data.