Sure! Let's fit a straight line trend to the given data using the method of least squares. We have been given the data as follows:
| Year | 2006 | 2007 | 2008 | 2009 | 2010 |
|------|------|------|------|------|------|
| Profit (in Rs '000) | 12 | 18 | 20 | 23 | 27 |
### Step 1: Calculate the Means
First, we need to calculate the mean of the years and the mean of the profits.
[tex]\[
\text{Mean of Years} = \frac{2006 + 2007 + 2008 + 2009 + 2010}{5} = 2008
\][/tex]
[tex]\[
\text{Mean of Profits} = \frac{12 + 18 + 20 + 23 + 27}{5} = 20
\][/tex]
### Step 2: Calculate the Coefficients [tex]\(b_0\)[/tex] and [tex]\(b_1\)[/tex]
We need to find the coefficients of the least squares regression line:
[tex]\[
\text{Profit} = b_0 + b_1 \times \text{Year}
\][/tex]
To find [tex]\(b_1\)[/tex]:
[tex]\[
b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\][/tex]
For the numerator of [tex]\(b_1\)[/tex]:
[tex]\[
\sum (x_i - \bar{x})(y_i - \bar{y}) = (2006 - 2008)(12 - 20) + (2007 - 2008)(18 - 20) + (2008 - 2008)(20 - 20) + (2009 - 2008)(23 - 20) + (2010 - 2008)(27 - 20)
\][/tex]
[tex]\[
= (-2)(-8) + (-1)(-2) + 0 + (1)(3) + (2)(7)
\][/tex]
[tex]\[
= 16 + 2 + 0 + 3 + 14 = 35
\][/tex]
For the denominator of [tex]\(b_1\)[/tex]:
[tex]\[
\sum (x_i - \bar{x})^2 = (2006 - 2008)^2 + (2007 - 2008)^2 + (2008 - 2008)^2 + (2009 - 2008)^2 + (2010 - 2008)^2
\][/tex]
[tex]\[
= (-2)^2 + (-1)^2 + 0 + 1^2 + 2^2
\][/tex]
[tex]\[
= 4 + 1 + 0 + 1 + 4 = 10
\][/tex]
So, [tex]\(b_1\)[/tex] is:
[tex]\[
b_1 = \frac{35}{10} = 3.5
\][/tex]
To find [tex]\(b_0\)[/tex]:
[tex]\[
b_0 = \bar{y} - b_1 \times \bar{x}
\][/tex]
[tex]\[
b_0 = 20 - 3.5 \times 2008
\][/tex]
[tex]\[
b_0 = 20 - 7028 = -7008
\][/tex]
### Step 3: The Estimated Straight Line
The regression equation is:
[tex]\[
\text{Profit} = -7008 + 3.5 \times \text{Year}
\][/tex]
### Step 4: Estimate the Profit for 2012
Substituting 2012 into the equation:
[tex]\[
\text{Profit}_{2012} = -7008 + 3.5 \times 2012
\][/tex]
[tex]\[
\text{Profit}_{2012} = -7008 + 7042
\][/tex]
[tex]\[
\text{Profit}_{2012} = 34
\][/tex]
### Step 5: Calculate the Trend Values
We now calculate the trend values for each given year:
[tex]\[
\text{Trend}_{2006} = -7008 + 3.5 \times 2006 = -7008 + 7021 = 13
\][/tex]
[tex]\[
\text{Trend}_{2007} = -7008 + 3.5 \times 2007 = -7008 + 7024.5 = 16.5
\][/tex]
[tex]\[
\text{Trend}_{2008} = -7008 + 3.5 \times 2008 = -7008 + 7028 = 20
\][/tex]
[tex]\[
\text{Trend}_{2009} = -7008 + 3.5 \times 2009 = -7008 + 7031.5 = 23.5
\][/tex]
[tex]\[
\text{Trend}_{2010} = -7008 + 3.5 \times 2010 = -7008 + 7035 = 27
\][/tex]
### Summary:
- The estimated profit for the year 2012 is 34 (in Rs '000).
- The trend values for the given years are:
- 2006: 13
- 2007: 16.5
- 2008: 20
- 2009: 23.5
- 2010: 27
