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 it 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 2,400 $ 10,900 February 2,500 $ 13,100 March 2,700 $ 11,700 April 3,000 $ 14,100 May 3,600 $ 16,100 June 5,000 $ 23,100 July 5,500 $ 22,100 August 5,500 $ 21,900 September 4,700 $ 22,700 October 4,000 $ 20,800 November 2,200 $ 10,600 December 2,800 $ 13,900 2. Using least-squares regression, estimate the variable cost per rental return and the monthly fixed cost incurred to wash cars. (Round Fixed cost to the nearest whole dollar amount and the Variable cost per unit to 2 decimal places.)

Answer 1

Ans: The solution is as follows :-

Firstly arrange the data in a tabular format -

Month	No. of cars	Total Cost
January	2400	10900
February	2500	13100
March	2700	11700
April	3000	14100
May	3600	16100
June	5000	23100
July	5500	22100
August	5500	21900
September	4700	22700
October	4000	20800
November	2200	10600
December	2800	13900

The formulas used here are as follows, and all the components will be worked out from the table made above.

n (Σxy) - (Σx)(Σy) Variable cost per unit = b =- n (Σx?)-(Σx)? (Σy)-b(Σx) Fixed cost = a=-- n

Table showing calculation of Variable cost per unit -

S.no.	x	y	x^2	xy
1	2400	10900	5760000	26160000
2	2500	13100	6250000	32750000
3	2700	11700	7290000	31590000
4	3000	14100	9000000	42300000
5	3600	16100	12960000	57960000
6	5000	23100	25000000	115500000
7	5500	22100	30250000	121550000
8	5500	21900	30250000	120450000
9	4700	22700	22090000	106690000
10	4000	20800	16000000	83200000
11	2200	10600	4840000	23320000
12	2800	13900	7840000	38920000
Total	43900	201000	177530000	800390000

Here,

n = 12

∑ x = number of cars = 43900

∑ y = Total cost = 201000

∑ x^2 = 177530000

∑ xy = 800390000

So after applying the given values in formula we get,

Variable cost per unit = 12*( 800390000) - (43900)*(201000)
12*(117530000) - (43900)^2
=(9604680000 - 8823900000 )
(2130360000 - 1927210000)
= (780780000)
(203150000)
= $ 3.84 / car

Now calculating Fixed cost -

Here,
∑ y = 201000
b = Variable cost per unit = $ 3.84
∑ x = 43900
n = 12
Applying the above values in formula we get,

Fixed cost = ( 201000) - (3.84*43900)
12
= (32424) / 12 = $ 2702