Question

1. Given the data,
x 1 2 3 4 5 6 7 8 9
y 9 8 10 12 11 13 14 16 15
(a) Calculate the coefficient of correlation?
(b) Obtain the line of regression.
(c) Estimate the value of y which should correspond to x = 6.2

Answer 1

Answer on Question #79272 – Math – Statistics and Probability

Question

Given the data:

x 1 2 3 4 5 6 7 8 9

y 9 8 10 12 11 13 14 16 15

(a) Calculate the coefficient of correlation

(b) Obtain the line of regression

(c) Estimate the value of $y$ which should correspond to $x = 6.2$

Solution

(a)

A correlation coefficient is given by a formula:

$$r_{xy} = \frac{\sum_{i=1}^{N} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{N} (x_i - \bar{x})^2 \sum_{i=1}^{N} (y_i - \bar{y})^2}},$$

where $x$ and $\bar{y}$ are the sample means of $x$ and $y$

Calculate the sample means (N=9):

$$\bar{x} = \frac{1}{N} \sum_{i=1}^{N} x_i = 5$$ $$\bar{y} = \frac{1}{N} \sum_{i=1}^{N} y_i = 12$$

Thus, the correlation coefficient $r_{xy} = 0.95$.

(b)

Consider the simple linear regression model and find the linear function $y = Ax + B$ that best fits the data. Use the method of least squares to find optimal values of parameters A and B.

It is necessary to minimize the sum:

$$S = \sum_{i=1}^{N} (y_i - (A x_i + B))^2$$

The minimum of the sum can be found by setting the partial derivatives to zero.

$$\frac{\partial S}{\partial A} = -2 \sum_{i=1}^{N} x_i (y_i - A x_i - B) = -2 N (\bar{x} \bar{y} - A \bar{x}^2 - B \bar{x}) = 0$$ $$\frac{\partial S}{\partial B} = -2 \sum_{i=1}^{N} (y_i - A x_i - B) = -2 N (\bar{y} - A \bar{x} - B) = 0,$$

where $\bar{x} \bar{y} = \frac{1}{N} \sum_{i=1}^{N} x_i y_i = \frac{199}{3}$, $\bar{x}^2 = \frac{1}{N} \sum_{i=1}^{N} x_i^2 = \frac{95}{3}$.

Thus, we receive the system of equations:

$$\left\{ \begin{array}{l} - A \overline {{x ^ {2}}} - B \bar {x} + \overline {{x y}} = 0 \\ - A \bar {x} - B + \bar {y} = 0 \end{array} \right.$$

Solving this system, we obtain the values of the parameters A and B:

$$A = \frac {\overline {{x y}} - \bar {x} \bar {y}}{\overline {{x ^ {2}}} - \bar {x}} = 0. 9 5$$ $$B = \frac {\overline {{x ^ {2}}} \bar {y} - \bar {x} \bar {x y}}{\overline {{x ^ {2}}} - \bar {x}} = 7, 2 5$$

The equation of the regression line is $y = 0.95x + 7.25$

(c)

To estimate the value of $y$ , which correspond to $x = 6.5$ we should substitute the value of $x$ into the equation of the line.

$$y = 0. 9 5 \times 6. 5 + 7. 2 5 = 1 3. 4 2 5$$

Comment

The data used in the calculations are given in table.

Answer:

(a) $r_{xy} = 0.95$

(b) $y = 0.95x + 7.25$

(c) $x = 6.5, y = 13.425$

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