Question

b) Following are demand and price data for 10 randomly selected samples. Here, x denotes demand, in units, and y denotes price, in hundreds of dollars. Predict the prices for X=22 and 23. (2 marks)
X 16 18 19 20 21 25 24 25 24 21
Y 290 280 295 425 384 315 355 328 325 425

Answer 1

1. Let us first determine $\sum x_i , \sum x^2_i , \sum y_i , \sum y^2_i , \sum x_iy_i$

$\sum x_i = 213 \\ \sum x^2_i = 4625 \\ \sum y_i = 3422 \\ \sum y^2_i = 1196690 \\ \sum x_iy_i = 73169$

n represents the sample size and thus n is equal to the number of ordered pairs.

n = 10

We can then determine the covariance using the formula

$s_{xy} = \frac{\sum x_iy_i - \frac{(\sum x_i)(\sum y_i)}{n}}{n -1} \\ = \frac{ 73169 - \frac{(213)(3422)}{10}}{10 -1} \\ = 31.15$

Let us next determine the sample variance s² using the formula:

$s^2 = \frac{\sum x^2_i - \frac{(\sum x_i)^2}{n}}{n-1} \\ s^2_x = \frac{ 4625 - \frac{(213)^2}{10}}{10-1} = 9.78 \\ s^2_y = \frac{ 1196690 - \frac{(3422)^2}{10}}{10-1} = 2568.16$

The sample standard deviation is the square root of the population sample:

$s_x = \sqrt{s^2_x} = \sqrt{9.78} = 3.12 \\ s_y = \sqrt{s^2_y} = \sqrt{2568.16} = 50.67$

We can then determine the correlation coefficient r using the formula:

$r = \frac{s_{xy}}{s_x s_y} \\ r = \frac{31.15}{3.12 \times 50.67} = 0.197$

2. The relationship between the variables is weak, positive, linear relationship.

3. Next, we can determine the slope b using the formula:

$b = r\frac{s_y}{s_x} \\ b = 0.197 \times \frac{50.67}{3.12} = 3.199$

Next, we can determine the y-intercept a using the formula $a = \bar{y} -b \bar{x}$ , where the sample mean is the sum of all values divided by the number of values.

$a = \bar{y} -b \bar{x} = \frac{\sum y_i}{n} -b \frac{\sum x_i}{n} \\ a = \frac{3422}{10} -3.199 \frac{213}{10} \\ = 342.2 -68.138 = 274.062$

Finally, we then obtain the regression line:

$y = a + bx = 274.062 + 3.199x \\ y(22) = 274.062 + 3.199 \times 22 = 344.44 \\ y(23) = 274.062 + 3.199 \times 23 = 347.64$