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Method of Least Squares, Goodness of Fit Deepa Dalal opened a free-standing radiology clinic. She had anticipated that the costs for the radiological tests would be primarily fixed, but she found that costs increased with the number of tests performed. Costs for this service over the past nine months are as follows:______.
Month Radiology Tests Total Cost
January 2,800 $133,500
February 2,600 135,060
March 3,100 175,000
April 3,500 170,600
May 3,400 176,900
June 3,700 186,600
July 3,840 174,450
August 4,100 195,510
September 3,450 185,300
Required:
1. Compute the cost formula for radiology services using the method of least squares. If required, round your answers to two decimal places.
Y = $ ________ + $ _________X
2. Using the formula computed in Requirement 1, what is the predicted cost of radiology services for October for 3,500 appointments? (Round the answer to the nearest dollar.)
$ _____________________

Respuesta :

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Answer:

Step-by-step explanation:

From the given information:

The cost formula for radiological services can be computed as:

Month      Radiology Test         Total Cost              

                              X                       Y                         XY

Jan              2800                      133500                   3738×10⁵

Feb              2600                      135060                  351156×10³

Mar              3100                       175000                   5425×10⁵

April             3500                      170600                   5971×10⁵

May              3400                      176900                  60146×10⁴

June             3700                      186600                  69042×10⁴

July              3840                       174450                  669888×10³

Aug              4100                        195510                   801591×10³

Sept             3450                       185300                 639285×10³  

                                                                                    52672×10⁵

Similarly, X²; =

Jan                  784×10⁴

Feb                   676×10⁴

Mar                  961×10⁴

April               1225×10⁴

May                1156×10⁴

June              1369×10⁴

July               14745600

Aug                16810000

Sept               11902500    

                      105168100

Now, suppose the equation is being represented by y = a+bx

where;

[tex]a = \dfrac{\sum Y * \sum X^2 -\sum X^2 \sum Y}{n * \sum X^2 - (\sum X)^2}[/tex]

[tex]a = \dfrac{1532920 *105168100-30490(52672\times 10^5)}{9 *105168100 - (30490)^2}[/tex]

a = 36588.821

Also,

[tex]b = \dfrac{n \sum XY - \sum X \sum Y}{n \sum X^2 -(\sum X)^2}[/tex]

[tex]b = \dfrac{9* 52672*10^5 -30990*1532920}{9*105168100-(30490)^2}[/tex]

b = 39.476

Thus;

Y = $36588.82 + $39.48X

The predicted cost of radiology for 3500 appointments will be :

Y = $36588.82 + $39.48X

Y = $36588.82 + $39.48(3500)

Y = 174768.82

Y ≅ $174769

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