Q.4.151 - Elementary Statistics

Question

Consider the following set of data points.

a. Compute the linear correlation coefficient, $r$.

b. Can you conclude from your answer in part (a) that the variables $x$ and $y$ are unrelated? Explain your answer.

c. Draw a scatterplot for the data.

d. Is use of the linear correlation coefficient as a descriptive measure for the data appropriate? Explain your answer.

e. Show that the data are related by the quadratic equation . Graph that equation and the data points.

Step-by-Step Solution

Answer

(a) The linear correlation coefficient, $r$ is $0$

(b)  $x$ and $y$ do not have a linear connection. Because the link between $x$ and $y$ could be nonlinear.

(c)

(d) No, Using the linear correlation coefficient as a descriptive measure for the given data is inappropriate. 

(e)

$\begin{array}{|lll|}\hline x& y& x^2\\ -3& 9& 9\\ -2& 4& 4\\ -1& 1& 1\\ 0& 0& 0\\ 1& 1& 1\\ 2& 4& 4\\ 3& 9& 9\\ \hline\end{array}$

Part (a)Step 1: Given information

Consider the following set of data points.

We need to compute the linear correlation coefficient, $r$ 

Part(a) Step 2: Explanation

The given data are,

$\begin{array}{|ll|}\hline x& y\\ -3& 9\\ -2& 4\\ -1& 1\\ 0& 0\\ 1& 1\\ 2& 4\\ 3& 9\\ \hline\end{array}$

The linear correlation coefficient is calculated as follows:

$r=\frac{\sum x_i{y}_i-\left(\sum x_i\right)\left(\sum y_i\right)/n}{\sqrt{\left.\mid \sum x_i^2-{\left(\sum x_i\right)}^2/n\right]\left|\sum y_i^2-{\left(\sum y_i\right)}^2/n\right|}},$

where the sample size is $n$.

As shown in the table below, the relevant sums can be determined.

$\begin{array}{|llllll|}\hline & x_i& y_i& X_i^2& Y_i^2& x_i{\chi }_i\\ & -3& 9& 9& 81& -27\\ & -2& 4& 4& 16& -8\\ & -1& 1& 1& 1& -1\\ & 0& 0& 0& 0& 0\\ & 1& 1& 1& 1& 1\\ & 2& 4& 4& 16& 8\\ & 3& 9& 9& 81& 27\\ \text{ Sum }& 0& 28& 28& 196& 0\\ \hline\end{array}$

Therefore,

$n=7$

$r=\frac{0-\left(0\right)\left(28\right)/7}{\sqrt{\left[0-(0{)}^2/7\right]\left[196-(28{)}^2/7\right]}}$

$=0$

Part(b) Step 1: Given information

Consider the following set of data points.

We need to find that from your answer in part (a) that the variables $x$ and $y$ are unrelated

Part (b) Step 2: Explanation

No, just the linear correlation between $x$ and $y$ is determined in portion (a), and the result is zero. As a result, the only conclusion is that $x$ and $y$ do not have a linear connection. Because the link between $x$ and $y$ could be nonlinear.

Part(c) Step 1: Given information

Consider the following set of data points.

We need to  draw a scatterplot for the data. 

Part(c) Step 2: Explanation

The scatterplot for the provided data is shown below.

Part(d) Step 1: Given information

Consider the following set of data points.

 We need to explain that whether the use of the linear correlation coefficient as a descriptive measure for the data appropriate

Part(d) Step 2: Explanation

No, the scatterplot has a considerable curvature and there is no linear relationship between $x$ and $y$. As a result, using the linear correlation coefficient as a descriptive measure for the given data is inappropriate.

Part(e) Step 1: Given information

Consider the following set of data points.

We need to show that the data are related by the quadratic equation $y=x^2$. And also need to graph that equation and the data points.

Part(e) Step 2: Explanation

Take a look at the table.

$\begin{array}{|lll|}\hline x& y& x^2\\ -3& 9& 9\\ -2& 4& 4\\ -1& 1& 1\\ 0& 0& 0\\ 1& 1& 1\\ 2& 4& 4\\ 3& 9& 9\\ \hline\end{array}$

It should be noticed that $y=x^2$. As a result, the data points satisfy the equation $y=x^2$.

The provided points and the equation $y=x^2$ are shown in the graph below.