Question

X 5 7 2 8 4
Y 8 12 6 14 8

a) Determine the extent to which output changes as the input change and interpret it. (3 marks)
b) Find the intercept of the output model and interpret it. Write down the equation of the line of best fit for X and Y. (3 marks)
c) Determine the value of the extent of relationship between input (X) and output (Y), and interpret your result. (3 marks)
d) Find the value of the coefficient of determination and interpret your result. (3 marks)
e) What will be the output level if the input usage is 15? (1 mark)
f) What percentage of the changes in output is due to other raw materials apart from X? (2 mark)
g) Test at 1 percent significance level, whether the correlation between X and Y exist. (5 marks)

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} X & Y \\ \hline 5 & 8 \\ \hline 7 & 12 \\ \hline 2 & 6 \\ \hline 8 & 14 \\ \hdashline 4 & 8 \end{array}$$

a) If the relationship between Y and X is believed to be linear, then the equation for a line may be appropriate: 

$$y=A+Bx$$

b)

$$mean=\bar{x}={\sum x_i \over n}={5+7+2+8+4 \over 5}=5.2$$

$$mean=\bar{y}={\sum y_i \over n}={8+12+6+14+8 \over 5}=9.6$$

$$S_{xx}=\sum (x_i-\bar{x})^2=\sum x_i^2-n\bar{x}^2=22.8$$

$$S_{yy}=\sum (y_i-\bar{y})^2=\sum y_i^2-n\bar{y}^2=43.2$$

$$S_{xy}=\sum (x_i-\bar{x})(y_i-\bar{y})=\sum x_iy_i-n\bar{x}\bar{y}=30.4$$

$$B={S_{xy} \over S_{xx}}={30.4 \over 22.8}={4 \over 3}\approx1.333333$$

$$A=\bar{y}-B\bar{x}=9.6-{4 \over 3}\cdot5.2={8 \over 3}\approx2.666667$$

$$y={8 \over3}+{4 \over 3}x$$

The intercept is the expected mean value of Y when all X=0.

When all inputs = 0, then the expected mean value of output will be $y={8 \over3}.$

c)

The slope is the rate of change, the mean amount of change in y when x increases by 1.  

The output y increases by 4/3, when x increases by 1.

d)

$$r^2={S_{xy}^2 \over S_{xx}S_{yy}}={30.4^2 \over 22.8\cdot43.2}\approx0.938272$$

The r² value tells us that 93.8% of the variation in the output is explained by the input.

e)

$$y={8 \over3}+{4 \over 3}(15)={68 \over3}\approx22.666667$$

f)

The $r^2$  value tells us that 93.8% of the variation in the output is explained by the input.

$$100\%-93.8\%=6.2\%$$

6.2 % of the changes in output is due to other raw materials apart from X.

g)

$H_0:$ The slope of the regression line is equal to zero.

$H_1:$ The slope of the regression line is not equal to zero.

$$SE=\sqrt{{S_{yy} \over (n-2)S_{xx}}}=\sqrt{{43.2 \over (5-2)22.8}}\approx0.7947$$

$$b_1={4 \over 3}\approx1.3333$$

$$df=n-2=5-2=3$$

$$t={b_1 \over SE}\approx{1.3333 \over 0.7947}\approx1.6778$$

Determine the p-value (two-tailed test)

$$p=2\cdot0.096=0.1920$$

Since the P-value (0.1920) is greater than the significance level (0.01), then the null hypothesis is not rejected.