Calculate the fixed and variable elements of the above mixed factory overhead using: i. The high and low method; ii. The linear regression analysis and determine the line of best fit.

From the table:

• Highest activity: July, 350 units, Overhead = 140
• Lowest activity: January and October, 150 units, Overhead = 80

The formula for variable cost per unit is:

$$\text{Variable Cost Per Unit} = \frac{\text{Highest Overhead Cost} - \text{Lowest Overhead Cost}}{\text{Highest Activity Level} - \text{Lowest Activity Level}}$$

Substitute the values:

$$\text{Variable Cost Per Unit} = \frac{140 - 80}{350 - 150} = \frac{60}{200} = 0.30$$

The variable cost per unit is $0.30.

Total fixed cost can be found by taking the total cost at either the high or low activity level and subtracting the total variable cost at that level.
Using the high activity level (July):

$$\text{Total Fixed Cost} = \text{Total Overhead at High Activity} - (\text{Variable Cost Per Unit} \times \text{High Activity Level})$$ $$\text{Total Fixed Cost} = 140 - (0.30 \times 350) = 140 - 105 = 35$$

Using the low activity level (January/October):

$$\text{Total Fixed Cost} = \text{Total Overhead at Low Activity} - (\text{Variable Cost Per Unit} \times \text{Low Activity Level})$$ $$\text{Total Fixed Cost} = 80 - (0.30 \times 150) = 80 - 45 = 35$$

The fixed cost is $35.

ii) The linear regression analysis and determine the line of best fit

We need the sum of $x$, $y$, $xy$, $x^2$, and the number of observations ($n$).

Month	Units (x)	Overhead (y)	$x^2$	$xy$
January	150	80	22500	12000
February	200	100	40000	20000
March	300	135	90000	40500
April	250	125	62500	31250
May	300	130	90000	39000
June	250	120	62500	30000
July	350	140	122500	49000
August	300	125	90000	37500
September	250	115	62500	28750
October	150	80	22500	12000
Total	2700	1150	660000	290000

$n = 10$

The formula for the slope ($b$) is:

$$b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}$$

Substitute the values:

$$b = \frac{10(290000) - (2700)(1150)}{10(660000) - (2700)^2}$$ $$b = \frac{2900000 - 3105000}{6600000 - 7290000}$$ $$b = \frac{-205000}{-690000} \approx 0.2971$$

The variable cost per unit (slope) is approximately $0.2971.

The formula for the y-intercept ($a$) is:

$$a = \frac{\sum y - b(\sum x)}{n}$$

Substitute the values:

$$a = \frac{1150 - 0.2971(2700)}{10}$$ $$a = \frac{1150 - 802.17}{10}$$ $$a = \frac{347.83}{10} \approx 34.783$$

The fixed cost (y-intercept) is approximately $34.78.

The line of best fit is represented by the equation $y = a + bx$, where $y$ is the total overhead, $a$ is the fixed cost, $b$ is the variable cost per unit, and $x$ is the number of units.

$$y = 34.78 + 0.2971x$$

Summary of results:

i) High and Low Method:

• Variable Cost Per Unit: $\boxed{\text{0.30}}$
• Fixed Cost: $\boxed{\text{35}}$

ii) Linear Regression Analysis:

• Variable Cost Per Unit: $\boxed{\text{0.2971}}$
• Fixed Cost: $\boxed{\text{34.78}}$
• Line of Best Fit: $\boxed{\text{y = 34.78 + 0.2971x}}$