Q.4.153 - Elementary Statistics

Question

Equation:$y=x^2$

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

b. Can you conclude from your answer in part (a) that the variables $x$ and y are linearly related? 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 equation provided.

f. Graph the equation and the data points.

Step-by-Step Solution

Answer

(a) Linear correlation coefficient  $r=0.9608$

(b) $x$ and $y$ can be said to be linearly connected.

(c)

   

(d) Using the linear correlation coefficient as a descriptive measure for the given data is inappropriate. 

(e)  

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

(f) 

Part (a)Step 1: Given information

Given in the question that,

$y=x^2$

We need to compute the linear correlation coefficient, .

Part(a) Step 2: Explanation

The given data are,

$\begin{array}{|ll|}\hline x& y\\ 0& 0\\ 1& 1\\ 2& 4\\ 3& 9\\ 4& 16\\ 5& 25\\ 6& 36\\ \hline\end{array}$

The equation is 

$y=x^2$

The linear correlation coefficient is calculated as follows:

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

where the sample size is $n$.

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

$\begin{array}{|llllll|}\hline & x_i& y_i& X_i^2& Y_i^2& x_i{y}_j\\ & 0& 0& 0& 0& 0\\ & 1& 1& 1& 1& 1\\ & 2& 4& 4& 16& 8\\ & 3& 9& 9& 81& 27\\ & 4& 16& 16& 256& 64\\ & 5& 25& 25& 625& 125\\ & 6& 36& 36& 1296& 216\\ \text{ Sum }& 21& 91& 91& 2275& 441\\ \hline\end{array}$

Therefore,

$n=7$

$r=\frac{441-\left(21\right)\left(91\right)/7}{\sqrt{\left[91-(21{)}^2/7\right]\left[2275-(91{)}^2/7\right]}}$

$=0.9608$

Part(b) Step 1: Given information

Given in the question that,

Equation:$\text{y }=x^2$

We need to find that from our answer in part (a) that the variables  and $y$ are linearly related .

Part (b) Step 2: Explanation

Yes, portion (a) calculates the linear correlation between $x$ and $y$, with a determined value of $0.9608$. The values are positively linearly connected if the computed r is close to$1$. As a result, $x$ and $y$ can be said to be linearly connected.

Part(c) Step 1: Given information

Given in the question that,

Equation:$y=x^2$

We need to  draw a scatterplot for the data. 

Part(c) Step 2: Explanation

For the provided data, the scatterplot is

Part(d) Step 1: Given information

Given in the question that,

Equation:$y=x^2$

 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 significant amount of curvature. As a result, using the linear correlation coefficient as a descriptive measure for the given data is inappropriate.

Part(e) Step 1: Given information

Given in the question that,

Equation:$\text{y }=x^2$

We need to show that the data are related by the  provided  equation. 

Part(e) Step 2: Explanation

Consider the table below.

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

$y=x^2$, it should be highlighted. As a result, the data points fulfil $y=x^2$.

Part(f) Step 1: Given information

Given in the question that,

Equation:$\text{y }=x^2$

We need to graph the equation and the data points .

Part(f) Step 2: Explanation

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