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Exercise 5A-2 Least-Squares Regression [LO5-11]

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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		$	11,100
February	2,500		$	13,200
March	2,700		$	11,900
April	3,000		$	14,400
May	3,600		$	16,300
June	5,100		$	23,500
July	5,500		$	22,300
August	5,500		$	22,400
September	4,700		$	22,900
October	4,000		$	21,400
November	2,200		$	10,800
December	2,800		$	14,600

Exercise 5A-2 Part 2

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.)

Fixed cost per month
Variable cost per rental return

Answer 1

Answer :-

First we calculate x² and xy and Sum of all x, y, x2 and xy, which gives us Σx, Σy, Σx2 and Σxy.

ху Month Rental Returns (x) Car Wash Costs (y) x2 January 2,400 11,100 February 2,500 13,200 March 2,700 11,900 April 3,000 14,400 May 3,600 16,300 June 5,100 23,500 July 5,500 22,300 August 5,500 22,400 September 4,700 22,900 October 4,000 21,400 November 2,200 10,800 December 2,800 14,600 5760000 6250000 7290000 9000000 12960000 26010000 30250000 30250000 22090000 16000000 4840000 7840000 26640000 33000000 32130000 43200000 58680000 119850000 122650000 123200000 107630000 85600000 23760000 40880000 Total 44000 204800 178540000 817220000

Σx= 44,000

Σy = 204,800

Σx² = 178,540,000

Σxy = 817,220,000

First we find variable cost per rental return -

Variable cost per rental return = N Σ(xy) − Σx Σy/ {N Σ(x²) − (Σx)²}

N = number of months = 12

Variable cost per rental return = { (12 × 817,220,000) - (44,000 × 204,800)}/ {(12 × 178,540,000) - (44,000)² }

Variable cost per rental return = (9,806,640,000 - 9,011,200,000)/(2,142,480,000 - 1,936,000,000)

Variable cost per rental return = 795,440,000 /206,480,000

Variable cost per rental return = $3.85

Now we calculated the Fixed Cost per month ,

Fixed Cost per month =( Σy - variable cost per rental return × Σx)/N

Fixed Cost per month = ( 204,800 - 3.85 × 44,000) /12

Fixed Cost per month = 35,400 / 12

Fixed Cost per month = $2,950