Question 1176480: 1. A manufacturer of basic commodities wants to know the relationship of advertising costs and sales. The data for a particular period on the two variables were recorded as follows:
Advertising costs (P1,000 )
20
40
30
50
25
30
40
35
45
20
25
40
Sales (P1,000)
385
400
489
580
410
475
510
500
525
360
420
480
Required:
a. Plot the data on a scatter diagram.
b. Find the equation of the regression line to predict sales from advertising costs.
c. Estimate the sales, if the advertising cost is P55,000.00.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's solve this problem step-by-step.
**a. Plot the Data on a Scatter Diagram**
* **X-axis:** Advertising Costs (P1,000)
* **Y-axis:** Sales (P1,000)
Plot the following points:
* (20, 385)
* (40, 400)
* (30, 489)
* (50, 580)
* (25, 410)
* (30, 475)
* (40, 510)
* (35, 500)
* (45, 525)
* (20, 360)
* (25, 420)
* (40, 480)
**b. Find the Equation of the Regression Line**
We need to find the equation of the form ŷ = a + bx, where:
* x = Advertising costs
* y = Sales
* b = slope
* a = y-intercept
First, calculate the necessary sums:
* Σx = 20 + 40 + 30 + 50 + 25 + 30 + 40 + 35 + 45 + 20 + 25 + 40 = 400
* Σy = 385 + 400 + 489 + 580 + 410 + 475 + 510 + 500 + 525 + 360 + 420 + 480 = 5434
* Σx² = 400 + 1600 + 900 + 2500 + 625 + 900 + 1600 + 1225 + 2025 + 400 + 625 + 1600 = 14300
* Σy² = 148225 + 160000 + 239121 + 336400 + 168100 + 225625 + 260100 + 250000 + 275625 + 129600 + 176400 + 230400 = 2499696
* Σxy = (20\*385) + (40\*400) + (30\*489) + (50\*580) + (25\*410) + (30\*475) + (40\*510) + (35\*500) + (45\*525) + (20\*360) + (25\*420) + (40\*480) = 7700 + 16000 + 14670 + 29000 + 10250 + 14250 + 20400 + 17500 + 23625 + 7200 + 10500 + 19200 = 190295
* n = 12 (number of data points)
Now, calculate b (slope):
* b = (nΣxy - ΣxΣy) / (nΣx² - (Σx)²)
* b = (12\*190295 - 400\*5434) / (12\*14300 - 400²)
* b = (2283540 - 2173600) / (171600 - 160000)
* b = 109940 / 11600
* b ≈ 9.4776
Next, calculate a (y-intercept):
* a = (Σy - bΣx) / n
* a = (5434 - 9.4776\*400) / 12
* a = (5434 - 3791.04) / 12
* a = 1642.96 / 12
* a ≈ 136.9133
Therefore, the regression equation is:
* ŷ = 136.9133 + 9.4776x
**c. Estimate the Sales if the Advertising Cost is P55,000.00**
* x = 55 (since the advertising cost is in P1,000)
* ŷ = 136.9133 + 9.4776\*55
* ŷ = 136.9133 + 521.268
* ŷ ≈ 658.1813
Therefore, the estimated sales are approximately P658,181.30.
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