The correlogram of the given data is show in Figure 4-3

The figure shows that the two curves move in the same direction and, moreover, they are very close to each other, suggesting a close relationship between price yield per plot (qtls) and quantity of fertilizer used (kg).

(a) The Scatter Diagram of the given data is shown in Figure

i. Figure shows that the plotted points are close to each other and reveal an upward trend.
ii. So there is a high degree of positive correlation between sales and expenses of the firm.

Broadly speaking, traditional or classical econometric methodology consists of the following steps.
1) Statement of the theory or hypothesis
2) Specification of the mathematical model of the theory
3) Specification of the econometric model of the theory
4) Obtaining the data
5) Estimation of the parameters of the econometric model
6) Hypothesis testing
7) Forecasting or prediction
8) Using the model for control or policy purposes.

he regression X on y is
\((X-\bar { X } )=r\times \frac { \sigma x }{ \sigma y } \times (Y-\bar { Y } )\)
Given x̅̅ = 12, \(\bar { Y } \) = 10
r = 0.85, σx = 0.1 and σy = 0.2
Then substituting the values in formula
(X-12) = 0.85 × (0.1/0.2) σy = 0.2
(X-12) = 0.85 × (0.5) × (Y-10)
X = 0.425 × (Y-10) + 12
X = 0.425y – 4.25 + 12
x = 0.425Y + 7.75
X on Y
Answer
X= 0.425Y + 7.75
The regression Y on X is
\((Y-\bar { Y } )=r\times \frac { \sigma x }{ \sigma y } \times (X-\bar { X } )\)
Given x̅ = 12, \(\bar { Y } \) = 10
r = 0.85, σx = 0.1 and σy =0.2
Then substituting the values in formula
(Y-10) = 0.85 × (0.2/0.1) × ( X–12)
(Y-10) = 0.85 × (2) × ( X–12)
Y = 1.7 × (X-12) + 10
Y = 1.7 X – 20.4 + 10
Y = 1.7 X + 10.4
Y on X
Y = 1.7 X – 10.4

Formula for Assumed Mean Deviation method.
\(r=\frac { N\sum { dxdy-(\sum { dx } )(\sum { dy } ) } }{ \sqrt { { N\sum { dx } }^{ 2 }-{ (\sum { dx } ) }^{ 2 } } \sqrt { { N\sum { dy } }^{ 2 }-{ (\sum { y } ) }^{ 2 } } } \)

\(\bar { X } =\frac { \sum { } }{ N } =\frac { 113 }{ 7 } =16\frac { 1 }{ 7 } \)
\(\bar { Y } =\frac { \sum { Y } }{ N } =\frac { 193 }{ 7 } =27\frac { 4 }{ 7 } \)
Take the assumed values A = 16 & B =27 therefore dx = X - A → X - 16 and Wdy = → Y - 27
\(r=\frac { N\sum { dxdy-(\sum { dx } )(\sum { dy } ) } }{ \sqrt { { N\sum { dx } }^{ 2 }-{ (\sum { dx } ) }^{ 2 } } \sqrt { { N\sum { dy } }^{ 2 }-{ (\sum { y } ) }^{ 2 } } } \)
\(=\frac { 7X187-1X4 }{ \sqrt { 7X159-{ (1) }^{ 2 } } \sqrt { 7X230-{ (4) }^{ 2 } } } \)
\(=\frac { 1309-4 }{ \sqrt { 1112 } \sqrt { 1610-16 } } \)
\(=\frac { 1305 }{ \sqrt { 1112 } \sqrt { 1594 } } =\frac { 1305 }{ \sqrt { 33.34 } \sqrt { 39.92 } } \)
\(=\frac { 1305 }{ 1330.9 } =0.9865\quad =0.986\)

\(r=\frac { \sum { xy } }{ \sqrt { { \sum { x } }^{ 2 }-\sum { { y }^{ 2 } } } } Where\quad X=(X-\bar { X } ),y=\sum { (y-\bar { y } ) } \)

Applying formula
\(r=\frac { \sum { xy } }{ \sqrt { { \sum { x } }^{ 2 } } \sqrt { \sum { { y }^{ 2 } } } } =\frac { 25 }{ \sqrt { 32 } \sqrt { 182 } } =\frac { 25 }{ 5.66x13.49 } =\frac { 25 }{ 76.35 } \)
r = 0.327, The car is getting old in years the cost of maintenane is also increasing. The age of Car and its maintenance are postively correlated.

Let us take price as X and supply as Y

\(r=\frac { N\sum { XY-(\sum { X } )(\sum { Y } ) } }{ \sqrt { N\sum { { X }^{ 2 }-{ (\sum { X } ) }^{ 2 } } \sqrt { N\sum { { Y }^{ 2 }-{ (\sum { y } ) }^{ 2 } } } } } \)
\(r=\frac { (5x3414)-70x239) }{ \sqrt { (5x1026)-{ (70 })^{ 2 } } \sqrt { 5x11685-{ (239) }^{ 2 } } } \)
\(r=\frac { 17,070-16,730 }{ \sqrt { 230x } \sqrt { 1304 } } \)
\(r=\frac { 340 }{ 547.65 } =+0.621\)
Pirce of the product and supply for the product is positively correlated. When price of the product increases then the supply for the product also incresases.

Procedure for Computing the Correlation Coefficient: (For Direct and Deviation from actual mean method).
Step- 1 Calculate the mean of two series ‘X’’Y’
Step- 2 Calculate the deviations ‘X’ and Y in two series from their respective mean.
Step- 3 Square each deviations of ‘X’ and ‘Y’ then obtain the sum of the Squared deviation,
Step- 4 Multiply each deviation under X with each deviation under Y and obtain the product of ‘xy’. Then obtain the sum of the product of X, Y. Then obtain the sum of the product of x,y is Σxy.
Step- 5 Substitute the value in the formula.

1. \(r=\frac { N\sum { XY-(\sum { X } )(\sum { Y } ) } }{ \sqrt { { N\sum { X } }^{ 2 }-{ (\sum { X } ) }^{ 2 } } \sqrt { { N\sum { Y } }^{ 2 }-{ (\sum { Y } ) }^{ 2 } } } \)
'r' is calculated by Direct Method without taking deviation of terms either from actual mean or assumed mean.
2. r is calculated by taking the Deviation from acutual mean.
\(r=\frac { \sum { xy } }{ { N\sum { X } }^{ 2 }-{ \sum { Y } }^{ 2 } }Where X=(X-\bar { X } ),Y=(\bar { X } )\)
3. 'r' is calculated by taking assumed mean
\(r=\frac { N\sum { dxdy-(\sum { dx } )(\sum { dy } ) } }{ \sqrt { { N\sum { dx } }^{ 2 }-{ (\sum { dx } ) }^{ 2 } } \sqrt { { N\sum { dy } }^{ 2 }-{ (\sum { dy } ) }^{ 2 } } } \)
Where dx refers to deviations of x series from assumed mean (x-x̅), dy refers to
deviations of y series from an assumed mean of (y-y)
1. Σdxdy = Sum of product of the deviations x and y series from their assumed means.
2. Σdx2 = Sum of the squares of the deviations of x series from an assumed mean
3. Σdy2= Sum of the squares of the deviations of y series from an assumed mean
4. Σdx = Sum of the deviation of x series from an assumed mean of x
5. Σdy = Sum of the deviation of y series from an assumed mean of y

1. Find out the actual mean of given data (x̅)
2. Find out the deviation of each value from the mean (x = X - x̅ )
3. Square the deviations and take the total of squared deviations Σx²
4. Divided the total Σx² by the number of observation \(\left( { \frac { \sum { x } }{ n } }^{ 2 } \right) \)
5. The square root of \(\left( { \frac { \sum { x } }{ n } }^{ 2 } \right) \) is standard deviation.

1. Statistics is not suitable to the study of qualitative phenomenon:
i. Since statistics is basically a science and deals with a set of numerical data.
ii. It is applicable to the study of quantitative measurements.
iii. As a matter of fact, qualitative aspects like empowerment, leadership, honesty, poverty, intelligence etc., cannot be expressed numerically and statistical analysis cannot be directly applied on these qualitative phenomena.
2. Statistical laws are not exact:
i. It is well known that mathematical and physical sciences are exact.
ii. But statistical laws are not exact and statistical laws are only approximations. Statistical conclusions are not universally true.
iii. They are true only on an average.
3. Statistics table may be misused:
i. Statistics must be used only by experts; otherwise, statistical methods are the most dangerous tools on the hands of the inexpert.
ii. The use of statistical tools by the inexperienced and untrained persons might lead to wrong conclusions.
4. Statistics is only one of the methods of studying a problem:
i. Statistical method does not provide complete solution of the problems because problems are to be studied taking the background of the countries culture, philosophy, religion etc., into consideration.
ii. Thus the statistical study should be supplemented by other evidences.

1. Statistics is not suitable to the study of qualitative phenomenon:
Since statistics is basically a science and deals with a set of numerical data. It is applicable to the study of quantitative measurements.
2. Statistical laws are not exact:
It is well known that mathematical and physical sciences are exact. But statistical laws are not exact and statistical laws are only approximations.
3. Statistics table may be misused:
Statistics must be used only by experts; otherwise, statistical methods are the most dangerous tools on the hands of the inexpert.
4. Statistics is only one of the methods of studying a problem:
Statistical method does not provide complete solution of the problems because problems are to be studied taking the background of the countries culture, philosophy, religion etc., into consideration.