Q. 4.152 - Elementary Statistics

Question

In each of Exercises 4.152 and 4.153. perform the following tasks.

 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 price data for

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 date points.

x	-5	-3	-1	1	3	5
y	-32	-60	-80	-92	-96	-92

Step-by-Step Solution

Answer

The output is 

Step 1. Given

x	-5	-3	-1	1	3	5
y	-32	-60	-80	-92	-96	-92

Step 2. Determine the correlation coefficient

MINITAB Procedure 

Step 1: Select Statistics> Basic Statistics> Connectivity.

Step 2: In the variable, select x and y in the box on the left.

Step 3: Click OK.

Step 3. MINITAB output

Correlation :x, y

Step 4. Solution b

Based on the result obtained in subsection (a), it cannot be concluded that the variables x and y are connected in a line. Picture information is also required whether the points are scattered on the line or not.

Step 5. Construct the scatterplot

Step 1: Select Graph> Scatterplot

Step 2: Select By Connection Line, then click OK.

Step 3: Under Yvariables, enter the y y column

Step 4: Below the X variable, enter the x column.

Step 5: Click OK.

Step 6. MINITAB Output

Step 7. Solution e

Show that the data is related to the given number

The data provided is related to the given number.

Description:

For x = - 5

Substitute -5 instead of x in the regression calculations. Therefore,

$$\begin{gathered} y=x^2-6x-8 \\ =(-5{)}^2-6(-5)-87 \\ =-32 \end{gathered}$$

$$\begin{gathered} For x=-3. \\ \mathrm{Substitute}-3 \mathrm{for} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{regression} \mathrm{equation}. \\ \mathrm{Therefore}, y=x²-6x-87 \\ =(-3)²-6(-3)-87 \\ =-60 \\ For x=-1₁ \\ \mathrm{Substitute} -1 \mathrm{for} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{regression} \mathrm{equation}. \\ \mathrm{Therefor}e, y=x²-6x-87 \\ =(-1)²-6(-1)-87 \\ =-80 \end{gathered}$$

Step 8. Explanation

$$\begin{gathered} For x=1. \\ \mathrm{Substitute} 1 \mathrm{for} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{regression} \mathrm{equation}. \\ \mathrm{Therefore}, y=x²-6.x-87 \\ =\left(1\right)-6\left(1\right)-87 \\ <=-92 \\ For x=3. \\ \mathrm{Substitute} 3 \mathrm{for} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{regression} \mathrm{equation}. \\ \mathrm{Therefore}, y=x²-6x-87 \\ =\left(3\right)\text{'}-6\left(3\right)-87 \\ =-96 \\ For x 5. \\ \mathrm{Substitute} 5 \mathrm{for} \mathrm{x} \mathrm{in} \mathrm{the} \mathrm{regression} \mathrm{equation} \\ \mathrm{Therefore}, y=x²-6x-87 \\ -\left(5\right)-6\left(5\right)-87 \\ = -92 \\ \mathrm{Therefore}, \mathrm{the} \mathrm{given} \mathrm{equation} \mathrm{y}=\mathrm{x}²-6\mathrm{x}-87 \mathrm{holds} \end{gathered}$$

Step 9. MINITAB Procedure

Step 1: Select Graph> Scatterplot

Step 2: Select By Connection Line, then click OK.

Step 3: Under Yvariables, enter the y y column. Step 4: Below the X variable, enter the x column.

Step 5: Click OK.

Step 10. MINITAB OUTPUT