Question

Bargain Rental Car offers rental cars in an off-airport location near a major tourist destination in California. Management would like to better understand the variable and fixed portions of its car washing costs. The company operates its own car wash facility in which each rental car that is returned is thoroughly cleaned before being released for rental to another customer. Management believes that the variable portion of its car washing costs relates to the number of rental returns. Accordingly, the following data have been compiled:

Month	Rental Returns		Car Wash Costs
January	2,500		$	11,700
February	2,500		$	13,500
March	2,800		$	12,500
April	3,100		$	15,300
May	3,700		$	16,900
June	5,200		$	24,700
July	5,600		$	22,900
August	5,700		$	23,900
September	4,800		$	23,500
October	4,500		$	23,200
November	2,300		$	11,400
December	3,100		$	16,700

Using least-squares regression, estimate the variable cost per rental return and the monthly fixed cost incurred to wash cars. (Round your Fixed cost to the nearest whole dollar amount and the Variable cost per unit to 2 decimal places.)

Answer 1

Step 1 Explaination about Cost

Cost

Cost is basically defined as expenditure incurred to produce a product or rendering of services. Again cost can be bifurcated into Fixed and variable cost  

Fixed Cost

Fixed cost are those cost which is incurred irrespective of the volume of increase/ decrease in value of goods / provision of services. This cost will be same up to specific point of level unless there is capacity constraint. examples are Rent , Depreciation etc.

Variable Cost

Variable cost are those cost which changes as per production, meaning by changes according to the volume of production , this are directly linked with production . Examples are direct material , direct labour etc.

Step 2 Explanation about Least-squares regression

Lease squares regression is working on statistical method to bifurcate the mixed / joint cost into fixed and variable cost based on regression analysis that reduces sume of squares error.

This line is basically represented in the form of equation : Y= a + bx

where variables defined as below

Y	Total Cost
a	Fixed Cost
b	Variable Cost per Rental Returns
X	Level of Activity /Units

Determination of Variable cost and Fixed Cost per unit as per below formula

Variable cost per unit = N Σ(xy) − Σx Σy / N Σ(x2) − (Σx)2

Fixed Cost per Unit = Σy − Varaibale Cost per unit * Σx/ N

where N is the no of months

Variable cost per unit = N Σ(xy) − Σx Σy / N Σ(x2) − (Σx)2

[(12*896060000)- (45800*216200)]/ [(12*192720000)- (45800*45800)] = 850760000/215000000

Variable Cost per unit  = $3.96 ( After rounding off up to two decimals )

Calculation of Fixed Cost

Fixed Cost  = Σy − Varaibale Cost per unit * Σx/ N

216200- 3.96*45800/12 = 216200- 15114

Fixed Cost     $ 201,086

Working Notes

Calculation of X , Y , X² and XY based on data

Month	Rental Returns ( X )	Car Wash Costs( Y)	X²	XY
January	2500	$11,700	6250000	$29,250,000
February	2500	$13,500	6250000	$33,750,000
March	2800	$12,500	7840000	$35,000,000
April	3100	$15,300	9610000	$47,430,000
May	3700	$16,900	13690000	$62,530,000
June	5200	$24,700	27040000	$128,440,000
July	5600	$22,900	31360000	$128,240,000
August	5700	$23,900	32490000	$136,230,000
September	4800	$23,500	23040000	$112,800,000
October	4500	$23,200	20250000	$104,400,000
November	2300	$11,400	5290000	$26,220,000
December	3100	$16,700	9610000	$51,770,000
Total	45800	216200	192720000	896060000