Question

PLEASE HELP! Use a graphing calculator to find an equation of the line of best fit for the data in the table. Round the slope and y-intercept to the nearest tenth. Then Identify and interpret the correlation coefficient.
X- 10.1 9.8 9.7 9.4 8.9 8.7 8.4 8.1
Y- 4.3 4.8 5.3 5.2 5.6 5.8 6.1 6.3
The equation of the line of best fit is y=
To the nearest thousandth, the correlation coefficient is r=

Answer 1

Answer:

ŷ = -0.895X + 13.600

Slope = -0.9

Intercept = 13.6

r = - 0.964

Step-by-step explanation:

Given the data:

X- 10.1 9.8 9.7 9.4 8.9 8.7 8.4 8.1

Y- 4.3 4.8 5.3 5.2 5.6 5.8 6.1 6.3

Inputting the data given into the online regression calculator :

The equation of line of best fit :

ŷ = -0.895X + 13.600

Slope = - 0.9(to the nearest tenth)

Intercept = 13.6 (to the nearest tenth)

The correlation Coefficient 'r' of the data supplied is - 0.9643, which implies that a strong negative correlation exists between the dependent and independent variables.

r = - 0.964 (to the nearest thousandth)

Answer 2

Inputting the data into the STAT then Fit Data function of the HP 50g

Graphing Calculator.

• The equation of the line of best fit is; y ≈ 13.6 - 0.9·x

• The correlation coefficient is r ≈ -0.964

The meaning of the strong negative correlation is that the x-values

of the data is decreasing, while the corresponding y-values of the data

increasing.

Reasons:

The least squares regression line equation is the best fit line for the data

The formula for the least squares regression line is $\hat Y = b \cdot X + a$

Where;

$b = \dfrac{n \cdot \sum X \cdot Y - \left (\sum X \right )\left (\sum Y \right )}{n \cdot \sum X^{2} - \left (\sum X \right )^{2}}$

$a = \dfrac{\sum Y - b \cdot \sum X}{n}$

n = The sample size

Inputting the given data into the function Fit Data on a graphing calculator,

by editing ∑DAT.

The left column, represent the x-values, and right column represent the y-

values as follows;

$\begin{array}{|c|cc|} \mathbf{Column \ 1}&&\mathbf{Column \ 2}\\10.1&&4.3\\9.8&&4.8\\9.7&&5.3\\9.4&&5.2\\8.9&&5.6\\8.7&&5.8\\8.4&&6.1\\8.1&&6.3\end{array}\right]$

After inputting the data as above, press enter.

Move the pointer to the Model options line, select choose to choose the

Linear Fit model by selecting OK.

Select OK again for the calculator to do the calculations.

The least squares regression line equation given by the calculator is

presented as follows;

y = 13.5998272884 - 0.894645941278·x

The slope and the y-intercept rounded to the nearest tenth gives;

• y = 13.6 - 0.9·x

• The slope = -0.9

• The y-intercept = 13.6

Correlation coefficient:

The correlation coefficient is given from the data using the following

formula;

$r = \dfrac{n \cdot \sum X \cdot Y - \left (\sum X \right ) \cdot \left (\sum Y \right )}{\sqrt{n \cdot \sum X^{2} - \left (\sum X \right )^{2}\times n \cdot \sum Y^{2} - \left (\sum Y \right )^{2}}}$

From the graphing calculator, we have the correlation coefficient, given as

follows;

Correlation: (-0.964276229529)

Therefore, to the nearest thousandth;

• The correlation coefficient, r ≈ -0.964

The high negative value of the correlation coefficient indicates a strong

and near perfect relationship between the variables, x, and y, such that we

have;

• A decrease in the value of x leads to a proportional increase in the value of y.

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