Q.4.152 - Elementary Statistics

Question

Equation: $y=x^2-6x-87$

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) The  linear correlation coefficient, $r=-0.8991$

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

(c) 

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

(e) 

$\begin{array}{|lll|}\hline x& y& x^2-6x-87\\ -5& -32& -32\\ -3& -60& -60\\ -1& -80& -80\\ 1& -92& -92\\ 3& -96& -96\\ 5& -92& -92\\ \hline\end{array}$

(f) 

Part (a)Step 1: Given information

Given in the question that,

Equation: $y=x^2-6x-87$

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

Part(a) Step 2: Explanation

The given data are,

$\begin{array}{|ll|}\hline x& y\\ -5& -32\\ -3& -60\\ -1& -80\\ 1& -92\\ 3& -96\\ 5& -92\\ \hline\end{array}$

The  Equation: $y=x^{2}-6 x-87$

The linear correlation coefficient is calculated as follows:

where the sample size is $n$.

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

$\begin{array}{|llllll|}\hline & x_i& y_{\mathrm{i}}& X_i^2& Y_i^2& x_i{X}_i\\ & -5& -32& 25& 1024& 160\\ & -3& -60& 9& 3600& 180\\ & -1& -80& 1& 6400& 80\\ & 1& -92& 1& 8464& -92\\ & 3& -96& 9& 9216& -288\\ & 5& -92& 25& 8464& -460\\ \text{ Sum }& 0& -452& 70& 37168& -420\\ \hline\end{array}$

Therefore,

$n=6$

$r=\frac{-420-\left(0\right)(-452)/6}{\sqrt{\left[70-(0{)}^2/6\right]\left[37168-(-452{)}^2/6\right]}}$

$=-0.8991$

Part(b) Step 1: Given information

Given in the question that,

 

Equation: $y=x^2-6x-87$

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

Part (b) Step 2: Explanation

The variables are negatively linearly connected if the estimated $r$ is close to $-1$.Yes, portion (a) calculates the linear correlation between $x$ and $y$, and the result is $-0.8991$. 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: $y=x^2-6x-87$

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

Given in the question that,

 

Equation: $y=x^2-6x-87$

 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: $y=x^2-6x-87$

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

Part(e) Step 2: Explanation

Observe the table below.

$\begin{array}{|lll|}\hline x& y& x^2-6x-87\\ -5& -32& -32\\ -3& -60& -60\\ -1& -80& -80\\ 1& -92& -92\\ 3& -96& -96\\ 5& -92& -92\\ \hline\end{array}$

$y=x^2-6 x-87$ is worth noting. As a result, the data points meet $y=x^2-6 x-87$.

Part(f) Step 1: Given information

Given in the question that,

Equation: $y=x^{2}-6 x-87$

We need to graph the equation and the data points .

Part(f) Step 2: Explanation

The provided points and the equation $y=x^2-6 x-87$ are represented in the graph below.